Implicit function theorem over free groups
Olga Kharlampovich
Department of Mathematics and Statistics
McGill University
Montreal, QC H3A 2K6, Canada
e-mail: [email protected]
Alexei Myasnikov
City College of New York
New York, NY 10031, USA
e-mail: [email protected]
August 30, 1999
1
In this paper we will prove a so-called Implicit function theorem for free groups
which is the analog of the corresponding theorem from Calculus. The main results are
formulated in Section 6.
1
Formulas over freely discriminated groups
Let G be a group generated by a set of generators A. By LA we denote the group theory
language with constants from A.
We say that a group G is freely discriminated if for every ¯nite subset K ½ G of
non-trivial elements of G there exists a free group F and a homomorphism Á : G ! F
such that g Á 6
= 1 for each g 2 K.
In this section we collect some results (old and new) on how to e®ectively rewrite
formulas over a non-Abelian freely discriminated group G into more simple or more
convenient "normal" forms. Some of these results hold in more general context, but we
do not present the most general formulations here, unless we need them in a due course.
Before we proceed with formulas we need to establish some properties of freely discriminated groups.
It is not hard to see that every freely discriminated group is a torsion-free CSA group
[2]. We recall from [29] that a group G is called a CSA group if every maximal Abelian
subgroup M of G is malnormal, i.e., M g \ M = 1 for any g 2 G ¡ M:
The class of CSA-groups is quite substantial. It includes all Abelian groups, all
torsion-free hyperbolic groups [29] and all groups acting freely on ¤-trees [39] as well
as many one-relator groups { a complete description of one-relator CSA-groups was
obtained by D. Gildenhuys, O. Kharlampovich and A. Myasnikov in [40].
Notice, that every CSA-group is commutative transitive [29]. A group G is called
commutative transitive if commutation is transitive on the set of all non-trivial elements
of G, i.e., if a; b; c 2 G ¡ f1g and [a; b] = 1; [b; c] = 1, then [a; c] = 1:
1.1
Atomic formulas over non-Abelian freely discriminated groups
The following result is due to A.Malcev [24] (he proved it for free groups, but his
argument works in a more general context). Namely, the argument applies to any
non-Abelian torsion-free commutative-transitive group G which satis¯es the so-called
Vought's conjecture:
(V) for any solution u = p; v = q; w = r of the equation u2 v2 w2 = 1 the elements p; q; r
pair-wise commute.
2
R.Lyndon [18] proved that every free group satis¯es (V). It implies that a freely discriminated group G also satis¯es (V) (otherwise one could discriminate a non-commuting
solution in G of the equation u2 v2 w2 = 1 into a free group - contradiction with the
Lyndon's result).
Lemma 1 Let G be a torsion-free non-Abelian commutative-transitive group satisfying the Vought's conjecture. Let a; b be arbitrary non-commuting elements in G. Then
the equation
x2ax2a¡1 = (ybyb¡1)2
(1)
has only the trivial solution in G, i.e. x = 1 and y = 1.
Proof. Let G be as above and let x; y be a solution in G of the equation (1) such
that x 6
= 1: Then
(x2a)2 a¡2 = ((yb)2 b¡2 )2:
(2)
In view of the condition (V), we deduce from (2) that [x2; a¡1 ] = 1 hence, by transitivity
of commutation, [x; a] = 1. Now, we can rewrite (2 in the form
x4 = ((yb)2 b¡2 )2;
which implies (according to (V)), that [x2 ; (yb)2 b¡2] = 1, and hence (since G is torsionfree)
x2 = (yb)2 b¡2 :
(3)
Again, it follows from (V) that [y; b] = 1. Henceforth, x2 = y2 and, by the argument
above, x = y. We proved that [x; a] = 1 and [x; b] = 1 therefore, by transitivity of
commutation, [a; b] = 1 - contradicting to the choice of a; b. This contradiction shows
that x = 1. In this event, the equation 2 transforms into
((yb)2b¡2)2 = 1;
which implies (yb)2 b¡2 = 1. Now from (V) we deduce that [yb; b] = 1, hence [y; b] = 1:
From (3) we see that y2 = 1, so y = 1, as desired.
Corollary 1 Let G be a non-Abelian freely discriminated group. Then for any ¯nite
system of equations S1 (x1 ; : : : ; xn ) = 1; : : : ; Sk (x1 ; : : : ; xn ) = 1 over G one can e®ectively
¯nd a single equation S(x1 ; : : : ; xn ) = 1 over G such that
VG (S1; : : : ; Sn ) = VG (S);
i.e., the formula
1.
Vk
i=1 (Si (x1 ; : : : ; xn )
= 1) is equivalent over G to the formula S(x1 ; : : : ; xn ) =
3
Proof. By induction it su±ces to prove the result for k = 2. In this case, by the lemma
above, the following equation
(S1(x1 ; : : : ; xn )2 a)2a¡2 = ((S2(x1 ; : : : ; xn )b)2 b¡2)2
can be chosen as the equation S = 1:
Moreover, observe that all the assumptions on the group G in Lemma 1 are ¯rst-order
assumptions, i.e., any group which is elementarily equivalent to the group G satis¯es
these assumptions. Now it follows from the proof of Lemma 1 that the following corollary
holds.
Corollary 2 Let G be a non-Abelian freely discriminated group. Then for any ¯nite
system of equations S1 (x1 ; : : : ; xn ) = 1; : : : ; Sk (x1 ; : : : ; xn ) = 1 over G one can e®ectively
¯nd a single equation S(x1 ; : : : ; xn ) = 1 over G such that
T hA (G) ` 8x1 : : : xn (
k
^
(Si (x1 ; : : : ; xn ) = 1) Ã! S(x1; : : : ; xn ) = 1):
i=1
The next lemma shows how to rewrite ¯nite disjunctions of equations into conjunctions of equations. In the case of free groups this result was known for years (in [25]
Makanin attributes this to Y.Gurevich). We give here a di®erent proof.
Lemma 2 Let G be a CSA group and let a; b be arbitrary non-commuting elements
in G. Then for any solution x; y 2 G of the system
[x; y a ] = 1; [x; y b ] = 1; [x; y ab ] = 1;
(4)
either x = 1 or y = 1. The converse is also true.
Proof. Suppose x; y are non-trivial elements from G, such that
[x; ya ] = 1; [x; y b ] = 1; [x; yab ] = 1:
Then by the transitivity of commutation [y b ; yab ] = 1 and [y a ; y b ] = 1. The ¯rst relation
implies that [y; ya ] = 1 and since a maximal Abelian subgroup M of G containing y is
malnormal in G, we have [y; a] = 1. Now from [y a; y b ] = 1 it follows that [y; y b ] = 1 and
consequently, [y; b] = 1. This implies [a; b] = 1, a contradiction, which completes the
proof of the lemma.
Combination of Lemmas 2 and 1 yields an algorithm to encode an arbitrary ¯nite
disjunction of equations into a single equation.
4
Corollary 3 Let G be a non-Abelian freely discriminated group. Then for any ¯nite
set of equations S1 = 1; : : : ; Sk = 1 over G one can e®ectively ¯nd a single equation
S = 1 over G such that
VG (S1) [ : : : [ VG (Sk ) = VG (S);
i.e., the formula (S1 = 1) _ : : : _ (Sk = 1) is equivalent over G to S = 1:
Inspection of the proof above shows that the following corollary holds.
Corollary 4 Let G be a non-Abelian freely discriminated group. Then for any ¯nite
set of equations S1 = 1; : : : ; Sk = 1 over G one can e®ectively ¯nd a single equation
S = 1 over G such that
T hA (G) ` 8x1 : : : xn (
k
_
i=1
Si = 1 Ã! S = 1):
Now, the corollaries above imply that one can e®ectively encode ¯nite conjunctions
and ¯nite disjunctions of inequalities over a non-Abelian freely discriminated group into
a single inequality.
Lemma 3 Let G be a non-Abelian freely discriminated group. Then for any ¯nite
set of inequalities S1 (x1 ; : : : ; xn ) 6
= 1; : : : ; Sk (x1 ; : : : ; xn ) 6
= 1 over G one can e®ectively
¯nd an inequality R(x1 ; : : : ; xn ) 6
= 1 and an inequality T (x1; : : : ; xn ) 6
= 1 over G such
that
T hA (G) ` 8x1 : : : xn (
k
^
(Si (x1 ; : : : ; xn ) 6
= 1) Ã! R(x1 ; : : : ; xn ) 6
= 1)
i=1
and
T hA (G) ` 8x1 : : : xn (
k
_
(Si (x1 ; : : : ; xn ) 6
= 1) Ã! T (x1 ; : : : ; xn ) 6
= 1):
i=1
Proof. By Corollary 4 there exists an equation R(x1 ; : : : ; xn ) = 1 such that
k
_
i=1
(Si (x1; : : : ; xn ) = 1) »G R(x1 ; : : : ; xn ) = 1:
(where by »G we denote the equivalence modulo T hA (G)). Hence
k
^
i=1
(Si 6
= 1) »G :(
k
_
i=1
(Si = 1) »G :(R = 1) » R 6
= 1:
5
This proves the ¯rst part of the result. Similarly,
k
_
(Si (x1 ; : : : ; xn ) 6
= 1) »G :(
i=1
k
^
(Si (x1; : : : ; xn ) = 1))
i=1
»G :(T (x1; : : : ; xn ) = 1) »G T (x1 ; : : : ; xn ) 6
=1
for some equation T (x1 ; : : : ; xn ) = 1 over G. This completes the proof.
Corollary 5 Let G be a non-Abelian freely discriminated group (in particular, a
non-Abelian free group). Then every atomic formula in the language LA is equivalent
over G to a formula of the type
n
_
i=1
1.2
(Si = 1 & Ti 6
= 1):
Positive sentences over F
In this section we re¯ne the Merzljakov's result on positive sentences over F [28]. In a
sense, we describe a complete set of de¯nable Skolem's functions for any given positive
sentence.
We remind here that for a group G and a set X by G[X] we denote the free G-group
with basis X, i.e., the free product G ¤ F (X).
We start with the following lemma.
Lemma 4 Let G be a non-Abelian freely discriminated group and U(X) = 1 be an
equation over G. Then
G j= 8X(U (X) = 1) () G[X] j= U(X) = 1;
i.e., only trivial equations have every tuple as a solution of it.
Proof. The group G[X] is freely discriminated [2]. Therefore, if the word U (X) is nontrivial element of G[X], then there exists a G-homomorphism Á : G[X] ! G, such that
UÁ 6
= 1: But then U (X Á ) 6
= 1 in G - contradiction with conditions of the lemma. So
U(X) = 1 in G[X]:
The next result shows how to eliminate quanti¯ers from positive universal formulas
over F .
6
Corollary 6 For a given word U(X; Y ) 2 F [X [ Y ] one can e®ectively ¯nd a word
W (Y ) 2 F [Y ] such that
F j= 8Y [8X(U(X; Y ) = 1)
$
W (Y ) = 1]:
(5)
Proof By the lemma above for any tuple of constants S from F the following equivalence
holds
F j= 8X(U (X; S) = 1) () F [X] j= U(X; S) = 1:
Now it su±ces to prove that for a given U(X; Y ) 2 F [X [ Y ] one can e®ectively ¯nd
a word W (Y ) 2 F [Y ] such that for any tuple of constants S over F the following
equivalence holds
F [X] j= U (X; S) = 1 () F j= W (S) = 1:
We do this by induction on the syllable length of U(X; Y ) which comes from the free
product F [X [ Y ] = F (X) ¤ F [Y ] (notice that F (X) does not contain constants from F ,
but F [Y ] does). If U(X; Y ) is of the syllable length 1, then either U (X; Y ) = U(X) 2
F (X) or U (X; Y ) = U(Y ) 2 F [Y ]. In the ¯rst event F j= U(X) = 1 means exactly that
the reduced form of U(X) is a trivial, so we can take W (Y ) trivial also. In the event
U(X; Y ) = U(Y ) we can take W (Y ) = U(Y ).
Suppose now that U (X; Y ) is of the syllable length more then 1, and in the free
product F (X) ¤ F [Y ] it has the following reduced form
U(X; Y ) = g1 (Y )v1 (X)g2(Y )v2 (X) : : : vm (X)gm+1 (Y )
where vi 's are reduced nontrivial words in F (X) and gi (Y )'s are reduced words in F [Y ]
which are all nontrivial except, possibly, g1(Y ) and gm+1(Y ).
If for a tuple of constants S over F we have F [X] j= U(X; S) = 1 then at least
one of the elements g2(S); : : : ; gm (S) must be trivial in F [Y ]. This observation leads
to the following construction. For each i = 2; : : : ; m delete the subword gi (Y ) from
U(X; Y ) and reduce the new word to the reduced form in the free product F (X) ¤ F [Y ].
Denote the resulting word by Ui (X; Y ). Notice that the syllable length of Ui (X; Y ) is
less then the length of U(X; Y ). It follows from the argument above that for any tuple
of constants S the following equivalence holds
F [X] j= U(X; S) = 1 () F [X] j=
m
_
(gi (S) = 1&Ui (X; S) = 1):
i=2
By induction one can e®ectively ¯nd words W2 (Y ); : : : ; Wm (Y ) 2 F [Y ] such that for
any tuple of constants S we have
F [X] j= Ui (X; S) = 1 () F j= Wi (S) = 1;
7
for each i = 2; : : : ; m. Combining the equivalences above we see that
F [X] j= U(X; S) = 1 () F j=
m
_
(gi (S) = 1&Wi (S) = 1):
i=2
By the corollaries from the previous section we can e®ectively rewrite the disjunction
m
_
(gi (Y ) = 1&Wi (Y ) = 1)
i=2
as a single equation W (Y ) = 1. That ¯nishes the proof.
Proposition 1 If
F j= 8X1 9Y1 : : : 8Xk 9Yk (u(X1 ; Y1 ; : : : ; Xk ; Yk ) = 1);
then there exist functions
Y1 = q1(X1); : : : ; Yk = qk (X1 ; : : : ; Xk );
such that
F [X1 ; : : : ; Xk ] j= u(X1 ; q1 (X1); : : : ; Xk ; qk (X1; : : : ; Xk )) = 1:
Proof.There is a ¯nite number of generalized equations corresponding to the equation
u = 1. Let Xi = fxij jj 2 Jg: Take xij = bamij1 bamij2 b : : : amijnij b such that all m's are
big and di®erent, and mtjk are chosen in such a way that Xt has nothing in common
except small (in comparison with mtjk ) powers of a with all Xi , Yi , for i < t. Some
of the generalized equations correspond to such a choice of X's. We choose x's so long
that each xij contains a subword bamijk b which is not cut by any boundary. Fix one
such subword for each xij . If this subword bamijk b belongs to a base ¸, then ¢¸ either
corresponds to the same subword of the same xij or to the subword of yts for t ¸ i.
Replace the choosen subwords bamijk b everywhere by bamijk x~ij b, where x~ij are new
symbols. The new words give solution of the same generalized equation.
Denote by xlij = bamij1 : : : bamijk and by xrij = b : : : amijnij b . Now substitute instead
of x~ij the word (xlij )¡1xij (xrij )¡1 .
We will now have a solution of the equation in group corresponding to our generalized
equation which gives the required result, because each yts will be expressed in terms of
xij ; i · t and constants. 2
8
1.3
Reduction to the implicit function form
Recall that we consider a free group F in the group theory language LA with constants
from the generating set A.
Proposition 2 For any ¯rst order sentence © in the language LA one can e®ectively
¯nd a sentence ª in the language LA which is of the form ª0 or :ª0 , where
ª0 = 9X1 8Y1 : : : 9Xk 8Yk (u(X1; Y1 ; : : : ; Xk ; Yk ) = 1 ! 9Zv(X1; Y1 ; : : : ; Xk ; Yk ; Z) = 1);
here u and vi are elements in the free group F [X1 ; Y1 ; : : : ; Xk ; Yk ; Z], such that
F j= © Ã! F j= ª:
Proof. For any sentence in the language LA one can e®ectively ¯nd its disjunctive
normal form (see, for example, [6]). By the results from the section 1.1 on atomic formulas in the language LA we can assume that the sentence © is already in the disjunctive
normal form in which every disjunct is a conjuction of an equality and an inequality, i.e.
© = 8X19Y1 : : : 8Xk 9Yk
r
_
(ui (X1 ; Y1 ; : : : ; Xk ; Yk ) = 1&vi (X1; Y1 ; : : : ; Xk ; Yk ) 6
= 1);
i=1
where X1 and Yk may be absent.
V. Diekert noticed that in the free group the formula
r
_
(ui (X) = 1&vi (X) 6
= 1)
i=1
is equivalent to
9z1 : : : zr
which is equivalent to
9z1 : : : zr (
r
_
i=1
r
_
(ui (X) = 1&vi (X) = zi &zi 6
= 1)
(ui (X) = 1&vi (X) = zi ))&ri=1 zi 6
= 1:
i=1
Wr
Since the system i=1 (ui (X) = 1&vi (X) = zi ) is equivalent to one equation, say
u(X; z1 ; : : : ; zr ) = 1, and the conjunction of inequalities &ri=1zi 6
= 1 is equivalent to one
inequality v(X; z1; : : : ; zr ) 6
= 1, © is equivalent to the formula
9
©1 = 8X1 : : : 8Xk 9Yk z1 ; : : : ; zr (u(X1 ; Y1 ; : : : ; Xk ; Yk ; z1; : : : zr ) = 1&
.
v(X1 ; Y1; : : : ; Xk ; Yk ; z1 ; : : : ; zr ) 6
= 1)
The negation of this formula has the form
©2 = 9X18Y1 : : : 9Xk 8Yk (u(X1 ; Y1 ; : : : ; Xk ; Yk ) = 1 ! v(X1 ; Y1 ; : : : ; Xk ; Yk ) = 1);
with new Yk . This formula satis¯es the conclusion of the proposition in the case when
Yk presents.
If Yk is absent we can write ©2 in the form
©3 = 9X18Y1 : : : 9Xk¡18Yk¡1 (8Xk u(X1 ; Y1 ; : : : ; Xk ) = 1 ! 9Xk v(X1 ; Y1; : : : ; Xk ) = 1):
By Corollary 6 one can e®ectively ¯nd a word w(X1 ; : : : ; Yk¡1 ; Xk ) such that the
equation w = 1 is equivalent over F to the formula 8Xk (u = 1): Replacing the latter
sub formula in ©3 by w = 1 we get the formula ª which satis¯es the conclusion of the
proposition. 2
2
Elements of algebraic geometry over groups
To formulate the implicit function theorem over groups we need to introduce some basic
notions of algebraic geometry over groups. We refer to [2] for details.
Let G be a group, F (X) be a free group with basis X = fx1 ; x2; : : : xn g, G[X] =
G ¤ F (X) be a free product of G and F (X). If S ½ G[X] then S = 1 is called a system
of equations over G. As an element of the free product the left side of every equation
in S = 1 can be written as a product of some elements from X [ X ¡1 (which are called
variables) and some elements from G (constants). To emphasize this we sometimes write
S(x1; : : : ; xn ) = 1 or S(X) = 1.
A solution of the system S(X) = 1 over a group G is a tuple of elements a1 ; : : : ; an 2
G such that after replacement of each xi by ai the left side of every equation in S = 1
turns into the trivial element of G. Equivalently, a solution of the system S = 1 over G
can be described as a homomorphism Á : G[X] ¡! G which is identical on G and such
that Á(S) = 1. By VG (S) we denote the set of all solutions in G of the system S = 1, it
is called the algebraic set de¯ned by S. This algebraic set VG (S) uniquely corresponds
to the normal subgroup
R(S) = fu(x) 2 G[X] j 8A 2 VG (S)u(A) = 1g
10
of the group G[X]. The subgroup R(S) contains S, and it is called the radical of S. The
quotient group
GR(S) = G[X]=R(S)
is the coordinate group of the variety V (S):
We de¯ne a Zariski topology on Gn by taking as a sub-basis for the closed sets of
this topology, the algebraic sets in Gn . If F is a free non-Abelian group then the union
of two algebraic sets is again algebraic, therefore the closed sets in the Zariski topology
over F are precisely the algebraic sets. The Zariski topology over F n is Noetherian for
every n, i.e., every proper descending chain of closed sets in F n is ¯nite. This implies
that every algebraic set V in F n is a ¯nite union of irreducible subsets (they are called
irreducible components of V ), and such decomposition of V is unique.
3
Algebraic sets over a free non-Abelian group
Let F be a free non-Abelian group. In this section, following [15] and [16], we describe
algebraic sets over F . Quadratic equations play the central part in this description.
A system S = 1 is called quadratic in variables X = fx1 ; : : : ; xn g if every variable
from X occurs in S not more then twice.
Let X1; : : : ; Xm be disjoint tuples of variables. A system U (X1 ; : : : ; Xm ) = 1 (with
coe±cients from F ) of the following type
S1(X1; X2; : : : ; Xm )
S2(X2; : : : ; Xm )
...
=1
=1
Sm (Xm ) = 1
is said to be triangular quasi-quadratic if for every i the system Si (Xi ; : : : ; Xm ) = 1 is
quadratic in the variables from Xi .
Denote by Gi the coordinate group of the subsystem Si = 1; : : : ; Sm = 1 of the
system U = 1:
Gi = F [Xi ; : : : Xm ]=R(Si (Xi ; : : : ; Xm ); : : : ; Sm (Xm )) (i = 1; : : : ; m + 1);
in particular, Gm+1 = F and G1 = FR(U) . The system U = 1 is said to be non-degenerate
(NTQ) if for each i the equation Si (Xi ; : : : ; Xm ) = 1 has a solution in Gi+1 (with
11
elements from Xi considered as variables and elements from Xi+1 ; : : : ; Xm as coe±cients
from Gi+1 ).
Observe, that if the system U = 1 is non-degenerate then the coordinate group Gi+1
is embeddable into Gi (i = 1; : : : ; m) ([15]), i.e., we have a chain of groups
F = Gm+1 · Gm · : : : · G1 = FR(U) :
To solve the system U = 1 over F one needs to solve the last quadratic equation
Sm (Xm ) = 1 over Gm+1 = F , then the previous one (which is again quadratic!)
Sm¡1(Xm¡1 ; Xm ) = 1 over the coordinate group Gm , and continue the process going
up along the triangular system until the ¯rst equation S1 (X1 ; : : : ; Xm ) = 1 has been
solved in the group G2. Now, to get solutions of this system in the initial free group
F , one needs to specialize the solutions obtained in G2 into F (in this case to specialize
means to take an arbitrary homomorphism from G2 into F , that ¯xes elements from F ,
and apply it to the obtained set of solutions in G2).
Now, the following crucial result from [16] describes the solution set in F of an
arbitrary system S(X) = 1 with coe±cients from F : for any such S(X) = 1 one
can e®ectively ¯nd a ¯nite family of non-degenerate triangular quasi-quadratic systems
U1 (Y1 ) = 1; : : : ; Un (Yn ) = 1 (here Yi 's are disjoint tuples of variables of, possibly, di®erent
length) and word mappings p1(Y1); : : : ; pn (Yn ) such that
VF (S) = p1 (VF (U1 )) [ : : : [ pn (VF (Un )):
The discussion above shows that algebraic sets de¯ned by quadratic equations are
building blocks for construction of arbitrary algebraic sets over F . This allows us to
focus now just on quadratic equations.
A standard quadratic equation over a group G is an equation of the one of the following
forms:
n
Y
[xi ; yi ] = 1;
n > 0;
(6)
i=1
n
Y
[xi ; yi ]
i=1
m
Y
zi¡1 ci zi d¡1 = 1;
i=1
n
Y
x2i = 1;
n; m ¸ 0; m + n ¸ 1;
(7)
n > 0;
(8)
i=1
n
Y
i=1
x2i
m
Y
zi¡1ci zi d¡1 = 1;
i=1
12
n; m ¸ 0; n + m ¸ 1;
(9)
where d; ci are nontrivial elements from G.
The equation S = 1 is strictly quadratic in variables X = fx1 ; : : : ; xn g if every letter
from X occurs in S exactly twice. A quadratic, not strictly quadratic, equation is easy
to solve over a group G (it has the form x = W (Y ), where the variable x does not occur
in W (Y )).
In the case when G is a free non-Abelian group a strictly quadratic equation over
G is equivalent to a standard one in the following way. Let S(X; Y ) = 1 be a strictly
quadratic equation in variables X over G. Then there is an automorphism Á of the free
group G ¤ F (X [ Y ) such that Á ¯xes all the letters from Y and all the elements from
G and such that Á(S) = 1 is a standard quadratic equation over G:
Thus the standard quadratic equations play a key part in constructing algebraic sets
over free groups. This explains the following de¯nition.
De¯nition 1 Let S = 1 be a standard quadratic equation over a free group F . Then
the algebraic set VF (S) is called an elementary neighborhood over F .
4
Schemes
Throughout the paper by capital letters X; Y; Z (maybe with indexes) we denote tuples
of variables.
The notion on a generalized equation (-) and the correspondent system of equations
¤
- in F can be found in [16], section 1.
Now we re¯ne the description of algebraic sets over free group F which was given in
the introduction. For this we need the following
De¯nition 2 A set of group words W in an alphabet A is called linear if every
symbol from A occurs at most in one word from W and in that word it occurs exactly
once (in the exponent §1) .
For example, if Z is an alphabet, then a set of words W = fw1 ; : : : ; wn g is linear over Z
if and only if there is a partition of the set Z:
Z = Z1 [ : : : [ Zn [ Z ¤;
§1
§1
where Zi = fzi1; : : : ; ziki g, such that wi = zi1
: : : zik
.
i
Now, for a triangular quasi-quadratic system
Q1 (Z1 ; : : : ; Zm )
13
=1
Q2 (Z2 ; : : : ; Zm )
=1
...
Qm (Zm ) = 1
(here Qi is a triangular quasi-quadratic system in Zi ) and m tuples of words
P1 = (p11 ; : : : ; p1k1 );
..
.
Pm = (pm1 ; : : : ; pmkm );
such that the tuple Pi is linear in the alphabet Zi we denote by Q[X1; : : : ; Xm j Z1 ; : : : ; Zm ] =
1 the following system of equations
Q1 (Z1 ; : : : ; Zm )
= 1;
Q2 (Z2 ; : : : ; Zm )
= 1;
...
Qm (Zm ) = 1;
x11 = p11 (Z1 ); : : : ; x1k1 = p1k1 (Z1);
x21 = p21 (Z2 ); : : : ; x2k2 = p2k2 (Z2);
..
.
xm1 = pm1 (Zm ); : : : ; xmkm = pmkm (Zm );
here X1 = (x11 ; : : : ; x1k1 ); : : : ; Xm = (xm1 ; : : : ; xmkm ).
This system Q[X1 ; : : : ; Xm j Z1; : : : ; Zm ] = 1 is called a scheme.
Let
V = VF (Q[X1 ; : : : ; Xm j Z1 ; : : : ; Zm ])
be the algebraic set over F de¯ned by the scheme. The projection of V onto the a±ne
space formed by the coordinates X1; : : : ; Xm is denoted by ShF (Q). Thus ShF (Q) is a
subset of F s , where s = jX1 j + : : : + jXm j, that consists of all tuples (C1; : : : ; Cm ) of
values of (X1 ; : : : ; Xm ) in F for which there are some tuples D1 ; : : : ; Dm of elements in
14
F such that Q[C1; : : : ; Cm j D1 ; : : : ; Dm ] = 1 in F . In other words, ShF (Q) is the truth
set in F of the Diophantine formula
9Z1 : : : 9Zm Q[X1; : : : ; Xm j Z1 ; : : : ; Zm ] = 1:
Notice, that the closure of ShF (Q) in the Zariski topology is an irreducible variety
[16].
Now we can formulate the result mentioned in the introduction in the following form.
Theorem 1 For every ¯nite system of equations S(X1; : : : ; Xm ) = 1 over a free
group F , one can ¯nd e®ectively a ¯nite family of schemes Ui [X1 ; : : : ; Xm jZ1 ; : : : ; Zm ] =
S
1 , such that VF (S) = i ShF (Ui ).
Proof. The proof of this theorem is just an inspection of the proof of the Theorem 2 in
[16]. The reader is assumed to be acquainted with the paper [16], but for convenience
we mention here several key steps of the proof. To construct this family of schemes we
¯rst have to consider all generalized equations for S = 1. Observe that when we replace
the initial system S = 1 by (¯nitely many) generalized equations, the variables xk 2 Xk
can be expressed as linear words in new variables hk (see the proof of Lemma 4 in [16].
Indeed, for a given xk we choose one occurence of x"k in S(X1 ; : : : ; Xm ) = 1 and then
xk = (h®k ; : : : ; h¯k ¡1 )" :
Then for each generalized equation - we begin the construction of the tree T (-)
announcing all closed sections of the interval I corresponding to variables from X1 as
working sections and the rest of the sections as strongly constant sections. Every strongly
constant section is considered as a constant section. We construct all Razborov's fundamental sequences corresponding to the tree T0 (-).
Let tp(v) = 2. Equation -v will be called ¯rst- level-nontrivial, if it has a closed
constant (but not strongly constant) section containing at least one base and not containing variables from the coe±cient equations. From the construction it follows that
-v is ¯rst-level-nontrivial if and only if the path from the initial vertex v0 to v contains
an auxiliary edge, corresponding to the case 15:1: If all the auxiliary edges correspond
to the cases 4,5, then the equation -v is ¯rst level trivial.
Let w be the terminal vertex of T0 (-); such that tp(w) = 2 and -w is ¯rst-levelnontrivial. Construct T3 (-) as in Section 8 in [16] and stop at the vertices w where
-w is ¯rst-level-trivial. Construct T4 (-) as in [16]. Then the ¯nal vertices of T4 (-)
will correspond to generalized equations on the interval corresponding to the variables
X2 ; : : : ; Xk . Then consider sections of the interval I corresponding to variables from X2
as working part and the rest of the sections as strongly constant part and iterate the
process.
15
Denote by T5 (-) the tree that we obtained. Each branch of T5 (-) corresponds to
one scheme.
Notice that all elementary transformations reexpress old variables as linear words
in some of new variables. It is enough to verify that transformation of variables corresponding to edges of the tree T4(-) also express old variables as linear words in some of
new variables.
Consider possible cases for edges of T4(-):
Consider ¯rst what happens if the edge of T4 (-) corresponds to the edge v ! w of
T1(-) and tp(v) = 7. Let hi+1 be a variable covered by a single base ¹: Suppose ¢(¹)
covers a section [h®(¢(¹)) ; : : : ; h¯(¢(¹))¡1 ]: In this event we delete the variable hi+1 from
the generatized equation -v , but we keep in among the variables for the scheme Uj ,
and we include equation hi+1 = h®(¢(¹)) ; : : : ; h¯(¢(¹))¡1 in the scheme. Notice that this
equation is quadratic in hi+1:
Consider now the edge of T4 (-) corresponding to the edge v ! w of T1(-) replacing
in¯nite path in T (-) corresponding to the cases 7 · tp(vk ) · 10 (see Lemma 7 and
Section 7 in [16]). In this case let variables hp1 ; : : : ; hpq be in the ker(-v ), variables
hi1 ; : : : ; hik be free variables (they correspond to variables from y¹ in Lemma 7), variables
hj1 ; : : : ; hjs be the rest of the variables not from the kernel. Then in the corresponding
scheme we have equations
hj1 = w1 (hj2 ; : : : ; hjs ; hp1 ; : : : ; hpq ; hi1 ; : : : ; hik );
hj2 = w2 (hj3 ; : : : ; hjs ; hp1 ; : : : ; hpq ; hi1 ; : : : ; hik );
:::
hjs = ws (hp1 ; : : : ; hpq ; hi1 ; : : : ; hik );
where w1 ; : : : ; ws are non-cancellable words in their variables. We keep these equations
in the scheme. Each of them is quadratic in the correspondent variable hjl , l = 1; : : : ; s:
We also keep empty quadratic equations in variables from fhi1 ; : : : ; hik g: We don't make
the back-track substitution in variables hj1 ; : : : ; hjs to get mappings pij 's. Those of the
variables hj1 ; : : : ; hjs ; hi1 ; : : : ; hik which appear in the expressions for initial variables,
appear there not more than once.
For the edges v ! w of T4 (-) corresponding to vertices v of type 14 and 15 the
quadratic system is system s1 = 1 de¯ned in the description of case 14. For the edges
v ! w of T4 (-) corresponding to vertices v of type 2 the quadratic system corresponds
to an extension of a centralizer.
16
Notice, that the linearity of the word mappings P1 ; : : : ; Pm in the de¯nition of scheme
is just formal, because not all variables from Zi appear in words from Pi , and variables
from Zi are sometimes expressed in terms of variables from Zi+1 ; : : : ; Zm :
2
It is clear from the proof above that every scheme Ui corresponding to the equation S(X1 ; : : : ; Xm ) = 1 comes from a speci¯c generalized equation which occurs in the
process described in the proof. Further we will need the precise form of these generalized equations. Let U = U [X1; : : : ; Xm jZ1; : : : ; Zm ] = 1 be one of the schemes that
correspond to the equation S(X1; : : : ; Xm ) = 1 from the theorem above. As we have
mentioned already U comes from a generalized equation -U which has the following
form. Suppose the scheme U[X1 ; : : : ; Xm j Z1; : : : ; Zm ] = 1 be given by the following
system of equations
Q1 (Z1 ; : : : ; Zm )
= 1;
Q2 (Z2 ; : : : ; Zm )
= 1;
...
Qm (Zm ) = 1;
x11 = p11 (Z1 ); : : : ; x1k1 = p1k1 (Z1);
x21 = p21 (Z2 ); : : : ; x2k2 = p2k2 (Z2);
..
.
xm1 = pm1 (Zm ); : : : ; xmkm = pmkm (Zm );
here X1 = (x11; : : : ; x1k1 ); : : : ; Xm = (xm1 ; : : : ; xmkm ). The interval I of the generalized
equation -U is partitioned into subintervals I1; I2; : : : ; Im which correspond to the tuples
of variables Z1 ; Z2 : : : ; Zm (these subintervals are situated on I consequently from the
left to the right, i.e., I1 is the leftmost on I, I2 is the second to the left and so on). Each
interval Ij is partitioned into qj subintervals Ijq , corresponding to the variables zjq in
the tuple Zj , 1 · q · qj , where qj = jZj j (again, these subinterval are situated on Ij
from the left to the right). The generalized equation -U is quadratic in the variables
z1q ; (1 · q · q1 ) which are exactly the items on the interval I1 , i.e., each item on I1
belongs exactly to two bases. Moreover, if we delete all the bases on the union I2 [: : :[Jm
which do not have the dual bases on the interval I1 the generalized equation obtained
after all these erasings and deletings corresponds to the system Q1(Z1; : : : ; Zm ) which is
quadratic in variables Z1 . Similarly, for an arbitrary interval Ij (1 < j) if we cut out
all the intervals to the left of Ij and consider only the union of intervals Ij [ : : : [ Im , and
17
also erase all the bases on Ij which have the dual bases somewhere to the left, i.e., on the
union I1 [ : : : [ Ij¡1 , then after such erasing each the item on Ij belongs exactly to two
bases, it means the new generalized equation is quadratic in the variables zjq , 1 · q · qj ,
which correspond to the items on Ij . Furthermore, if erase all the bases on the union
Ij+1 [: : :[Im which do not have the dual bases on the interval Ij the resulting generalized
equation determines the quadratic system Qj (Zj ; : : : ; Zm ) = 1. Since U is a scheme then
we can partition each tuple of variables Zj into disjoint union Zj1 [ : : : [ Zjkj [ Zj¤ such
that for each variable xjk from the tuple Xj we have xjk = pjk (Zjk ) (k = 1; : : : ; kj ): We
can view Xj as situated on the interval Ij starting from the left and covering all the items
from Zj1 [: : :[Zjkj , so the interval Ij is a disjoint union of Xj and Zj¤ . Moreover, we can
assume that for each base on Zj¤ that has the dual base on Xj this dual base is an item
from Xj . We have described the generalized equation -U . Notice, that the fact that
the scheme U comes from the equation -U means, ¯rstly, that all solutions of the group
equation -¤U are solutions of the triangular quasi-quadratic system Q1 = 1; : : : ; Qi = 1
and vice versa; and, secondly, that the Diophantine set ShF (U) can be obtained from
solutions of -¤U by means of the linear words P1; : : : ; Pm as was described above.
We de¯ne a partition of an interval (or of a variable) as a disjoint sequence of bases
and items, such that their union is the whole interval. If we add ¯ctitious bases instead
of items, we can say that a partition is a disjoint union of bases. Say x = ¸i11 : : : ¸i1ki1 =
: : : = ¸is1 : : : ¸iskis :
De¯nition 3 By a cutting corresponding to the equation S(X1; : : : ; Xm ) = 1 we call
a generalized equation - which consists of a union of partitions for the variables from
X1 ; : : : ; Xm and such that any solution of - being substituted into X1 ; : : : ; Xm , gives a
solution of S = 1.
Theorem 10 For every ¯nite system of equations S(X1 ; : : : ; Xm ) = 1 over a free group
F and any j, such that 1 · j · m; one can ¯nd e®ectively a ¯nite family of generalized
equations -i (Yi ) = 1, such that each -i is a cutting of variables from Xj ; : : : ; Xm with
the property that the variety V (-¤i ) is a union of projections of ShF (Ui ) and ShF (Uk )
for some other k's from Theorem 1 into variables from Yi . In addition, the system of
equations Q1 = 1; : : : ; Qj¡1 = 1 from the scheme Ui = 1 can be written as a system with
coe±cients from Yi : Q1 (Z1 ; : : : ; Zj¡1 ; Yi ) = 1; : : : ; Qj¡1 (Zj¡1; Yi ) = 1 and has a solution
in FR(-¤i ) :
Proof
The proof is again an inspection of the proof of Theorem 2 from [16] with re¯ning
of some results from Section 6 in [16]. For example, the cuttings of X1; : : : ; Xm can be
easily obtained from the generalized equations on the ¯rst step of the reduction process.
18
We need the following result to be able to treat the case of a singular periodic
structure in a way di®erent from paper [16].
Lemma 5 Let - be a consistent generalized equation with no boundary connections,
periodic with respect to the singular periodic structure < P; R > .
There exists a solution H + of - with the property that for any hk 2 P such that Hk =
+
P2 P nk P1 (P2 and P1 are an end and a beginning of P ) Hk+ = P2 P nk P1, where nk ; n+
k > 0
+
and the numbers nk 's are bounded by a certain computable function f2(-; P; R). For all
hk 6
2 P Hk = Hk+ :
Proof
We use he notation from Section 6, paper [16]. Let ±((k)) = P1(k) P2(k) . Denote by
t(¹; hk ) the number of occurrences of the edge with label hk in the cycle c¹ , calculated
taking into account the orientation. Let
Hk = P2(k) P nk P1(k+1)
(10)
(hk lies on a closed section from P), where this equality is graphic whenever hk 2 P.
Direct calculations show that
H(b¹ ) = P
P
k
t(¹;hk )(nk +1)
:
(11)
This equation implies that the vector fnk g is a solution to the following system of
Diophantine equations in variables fzk jhk 2 Pg:
§hk 2P t(¹; hk )zk + §hk 6
2P t(¹; hk )nk = 0;
¹ 2 P:
Note that the number of unknowns and coe±cients of this system are bounded from
above (jnk j · 2 for hk 6
2 P) by a certain computable function of -; P; and R.
A solution fmk g of a system of linear Diophantine equations is called minimal , if
+
mk ¸ 0 and there is no other solution fm+
k g such that 0 · mk · mk for all k, and that
at least one of the inequalities m+
k · mk is strict.
Take as n+
for
h
2
P
some
minimal
solution of the system. 2
k
k
To construct the collection of cuttings, we ¯rst have to consider all generalized equations for S = 1. Then for each generalized equation - we begin the construction of
the tree T (-) announcing all closed sections of the interval I corresponding to variables
from X1 as working sections and the rest of the sections as strongly constant sections.
19
We construct all Razborov's fundamental sequences corresponding to the tree T0(-) in
a di®erent way which we will now describe.
This lemma allows us, while constructing the tree T0 (-); to deal with singular periodic structures the same way as with regular periodic structures. Namely, if in some
vertex v in the construction of T0 (-) the leading base overlaps with its double, we replace the solution by another solution minimal with respect to all canonical groups of
automorphisms for all regular periodic structures and minimal in a sense of Lemma 5 for
singular periodic structures corresponding to vertices of type 2 in T (-) situated above
v. Therefore, the ¯nal vertices w of T0 (-) will only have type 1, and at these vertices
constructing T0 (-w ) we will continue working with the same generalized equation -w ,
until it is changed so that the working part is empty. The automorphisms that we used
to change solutions don't change the variables from the constant part. Hence at the
end of the process, when the working part is empty, we have a cutting of variables from
X2 ; : : : ; Xm : Then we continue working with this cutting and so on.
To prove the last statement of the Theorem about system Q1 = 1; : : : ; Qj¡1 = 1
we summarize some results from Section 6 in [16] in the following lemma. Notice, that
without loss of generality we can assume that v0 corresponds to the beginning of the
period P .
Lemma 6 Let - be a consistent generalized equation periodic with respect to a periodic structure < P; R >. Then the following is true.
¹ of - is minimal with respect to the trivial group of automorphisms
1. If the solution H
of FR(-¤ ) then there is a cycle c in Z~ such that H(c) = P n0 , and n0 < 2½:
2. Let ¹a; x¹ be the generators of the group FR(-¤ ) constructed in [16]. If ei 2 P \ T ,
and c is a cycle beginning at the initial vertex of ei , then the mapping de¯ned as h(ei ) !
h(c)k h(ei ) (k is any integer) on the generator h(ei ) and ¯xing all the other generators
can be extended to the automorphism of FR(-¤ ) :
3. If ei 2 P n T , c and ce have the same initial vertex, then the mapping de¯ned as
h(ce ) ! h(c)k h(ce ) and ¯xing all the other generators can be extended to the automorphism of FR(-¤ ) :
Notice, that the proof of the ¯rst two parts is contained in the proof of Lemma 14
in [16] and does not use the regularity of the structure in Lemma 14.
+
(k)
(k)
From Lemma 5 it follows that (P2 )¡1Hk+ Hk¡1 P2 = P nk ¡nk commutes with H + (c),
hence for hk = h(ei ); ei 2 P \ T the value Hk can be obtained by a composition
of an automorphism of type 2 (Lemma 6) and a suitable bounded solution H + of -.
After obtaining this way all Hk 's for hk = h(ei ); ei 2 P \ T , we can obtain Hk 's for
hk = h(ei ); ei 2 P n T using automorphisms of type 3.
20
The theorem is proved.
A scheme together with the corresponding cuttings of Xj ; : : : ; Xm for all j; 1 · j · m
is called a generalized scheme.
5
De¯nitions and elementary properties of liftings
We need few preliminary lemmas before we give the de¯nition of the lifting. Recall that
by de¯nition if S(X) = 1 is an equation over G, then
GS = G[X]=ncl(S);
Rad(S) =
\
fker Á j Á : GS ! Gg;
GR(S) = G[X]=Rad(S):
More generally, for a G-group H we denote by R(H) the intersection of the kernels
of all G-homomorphisms from H onto G (if there are no such homomorphisms, then
R(H) = H). De¯ne H ¤ = H=R(H), then H ¤ is a G-group (provided, H ¤ 6
= 1) which is
G-separated by G. Notice, that (GS )¤ = GR(S).
Lemma 7 Let ® : H1 ! H2 be a G-homomorphism and suppose © = fÁ : H2 ! Gg
be a G-separating family of G- homomorphisms. Then
ker ® =
\
fker(® ± Á) j Á 2 ©g
Proof. Suppose h 2 H1 and h 6
2 ker(®): Then ®(h) 6
= 1 in H2. Hence there exists Á 2 ©
such that Á(®(h)) 6
= 1. 2
Corollary 7 Let H1 and H2 be G-groups. Then:
1. If H2 is G-separated by G, then R(H1 ) µ ker(®) for each G-homomorphism
H1 ! H2 ;
®:
2. Every G homomorphism Á : H1 ! H2 gives rise to a G-homomorphism Á¤ : H1¤ !
H2¤ , provided the groups H1¤ and H2¤ are both non-trivial.
Proof. 1) We have
R(H1 ) =
\
fker Á j Á : H1 ! Gg µ
\
21
fker(® ± ¯) j ¯ : H2 ! Gg = ker ®:
2) Let ® : H1 ! H2¤ be composition of the following homomorphisms
Á
´
H1 ! H2 ! H2¤ :
Then by 1) R(H1 ) µ ker ®, therefore ® induces the canonical G-homomorphism Á¤ :
H1¤ ! H2¤. 2
Let S(X) = 1 be an equation over G, denote by ¸ : G ! GS and ¹ : G ! GR(S)
the canonical homomorphisms induced by the inclusion G ,! G[X] and then by the
factorization to the corresponding quotient group.
Lemma 8
1. ¸ : G ! GS is an embedding () S(X) = 1 has a solution in some
G-group H;
2. ¹ : G ! GR(S) is an embedding () S(X) = 1 has a solution in some G-group H
which is G-separated by G.
Proof. 1) If S(x1 ; : : : ; xm ) = 1 has a solution (h1 ; : : : ; hm ) in some G-group H, then
the G-homomorphism xi ! hi ; (i = 1; : : : ; m) from G[x1 ; : : : ; xm ] into H induces a
homomorphism Á : GS ! H, since H is a G-group all non-trivial elements from G are
also non-trivial in the factor-group GS , therefore ¸ : G ! GS is an embedding.
2) Let S(x1 ; : : : ; xm ) = 1 has a solution (h1 ; : : : ; hm ) in some G-group H which is
G-separated by G. Then there exists the canonical G-homomorphism ® : GS ! H
de¯ned as in 1). Hence Rad(S) µ ker ® by the corollary above, and then ® induces
a homomorphism from GR(S) into H which is monic on G. Therefore, G embeds into
GR(S) . 2
Now we can apply Corollary 7 to coordinate groups of nonempty algebraic sets.
Lemma 9 Let subsets S and T from G[X] de¯ne non-empty algebraic sets in a
group G. Then every G-homomorphism Á : GS ! GT gives rise to a G-homomorphism
Á¤ : GR(S) ! GR(T ) .
Proof. The result follows from Corollary 7 and the lemma above.
Now we are in a position to give the following
De¯nition 4 Let S(X) = 1 be an equation over a group G which has a solution in
G. We say that an equation T (X; Y ) = 1 is compatible with S(X) = 1 over G if for
every solution ¹a of S(X) = 1 in G the equation T (¹
a; Y ) = 1 also has a solution in G,
i.e., the algebraic set VG (S) is a projection of the algebraic set VG (S; T ):
22
The next proposition describes compatibility of two equations in terms of their coordinate groups.
Proposition 3 Let S(X) = 1 be an equation over a group G which has a solution
in G. Then T (X; Y ) = 1 is compatible with S(X) = 1 over G if and only if GR(S) is
canonically embedded into GR(S;T ); and every G- homomorphism ® : GR(S) ! G extends
to a G- homomorphisms ®0 : GR(S;T ) ! G.
Proof. Suppose ¯rst, that T (X; Y ) = 1 is compatible with S(X) = 1 over G and suppose
that VG (S) 6
= ;: The identity map X ! X gives rise to a G-homomorphism
¸ : GS ¡! GS;T
(notice that both GS and GS;T are G-groups by Lemma 8) which by Lemma 9 induces
a G-homomorphism
¸¤ : GR(S) ¡! GR(S;T ) :
We claim that ¸¤ is an embedding. To this end we need to prove the second statement about the extensions of homomorphisms. Let ® : GR(S) ! G be an arbitrary
G-homomorphism. It follows that a¹ = ®(X) is a solution of S(X) = 1 in G. Since
T (X; Y ) = 1 is compatible with S(X) = 1 over G there exists a solution, say ¹b; of
T (¹
a; Y ) = 1 in G. A map
X ! ®(X); Y ! ¹b
gives rise to a G-homomorphism G[X; Y ] ! G, which induces a G-homomorphism
Á : GS;T ! G. By Lemma 9 Á gives rise to a G-homomorphism
Á¤ : GR(S;T ) ¡! G:
Clearly, Á¤ makes the following diagram to commute.
¸¤
GR(S)
¡
®
? ¡ª
¡
¡
¡
¡ Á¤
G
23
¡
¡
GR(S;T )
Now we ¯nish the proof that ¸ is an embedding. Observe, that GR(S) is G-separated by G,
therefore for every non- trivial g 2 GR(S) there exists a G-homomorphism ® : GR(S) ! G
such that ®(G) 6
= 1. But then Á¤ (g) 6
= 1 and consequently g 6
2 ker ¸.
The inverse statement is obvious. 2
Let S(X) = 1 be an equation over G and suppose VG (S) 6
= ;. The canonical
embedding X ! G[x] induces a map
¹ : X ! GR(S):
We are ready to formulate the main de¯nition.
De¯nition 5 Let T (X; Y ) = 1 is compatible with S(X) = 1 over G. We say that
T (X; Y ) = 1 admits a lift to a generic point of S = 1 over G (or shortly S-lift over G) if
T (X ¹ ; Y ) = 1 has a solution in GR(S) (here Y are variables and X ¹ are constants from
GR(S) ).
The next result characterizes liftings in terms of the coordinate groups of the corresponding equations.
Proposition 4 Let S(X) = 1 be an equation over G and suppose that VG (S) 6
=
;. Then for an arbitrary equation T (X; Y ) = 1 over G the following conditions are
equivalent:
1. T (X; Y ) = 1 admits S-lift over G;
2. GR(S;T ) 'GR(S) (GR(S) )R(T (X ¹ ;Y )) ;
3. there exists a GR(S) -homomorphism GR(S;T ) ! GR(S) (by Proposition 3 we view
GR(S) as a subgroup of GR(S;T ) ).
Proof. 1) =) 2). The identity maps X ! X; Y ! Y give rise to a G-homomorphism
GS;T ¡! (GR(S) ¤ F (Y ))=ncl(T (X ¹ ; Y ))
which by Lemma 9 induces a G-homomorphism
± : GR(S;T ) ¡! (GR(S) )R(T (X ¹;Y ) :
Notice, that we used here the condition that T (X ¹ ; Y ) = 1 has a solution in GR(S) :
24
Now we construct the inverse of ±. The same identity maps X ! X; Y ! Y give
rise to a G-homomorphism
¾1 : GR(S) [Y ] ¡! GR(S;T ):
Indeed, it is su±ce to notice that RadG (S) µ RadG (S; T ): Obviously, ¾1 induces the
canonical homomorphism
¾2 : GR(S) [Y ]=ncl(T (X ¹ ; Y )) ¡! GR(S;T ):
By Lemma 9 ¾2 gives rise to a homomorphism
¾ : (GR(S) )R(T (X ¹ ;Y )) ¡! GR(S;T ) ;
which is an embedding on GR(S) (see the proposition above). Notice, that ¾ ± ± and ± ± ¾
are identical on X; Y; G. Since the groups GR(S;T ) and (GR(S) )R(T (X ¹ ;Y )) are generated by
X; Y; G then both the homomorphisms ¾ ± ± and ± ± ¾ are identical, and , consequently,
both ¾ and ± are isomorphisms.
The implications 2) =) 3) and 3) =) 1) are obvious. 2
One can ask whether it is possible to lift a system of equations and inequalities into a
generic point of some equation S = 1? This is the question that we are going to address
below. We start with very general de¯nitions.
De¯nition 6 Let S(X) = 1 be an equation over a group G which has a solution in
G. We say that a formula ©(X; Y ) in the language LA is compatible with S(X) = 1
over G if for every solution a¹ of S(X) = 1 in G there exists a tuple ¹b over G such that
the formula ©(¹a; ¹b) is true in G, i.e., the algebraic set VG (S) is a projection of the truth
set of the formula ©(X; Y ) & (S(X) = 1):
De¯nition 7 Let a formula ©(X; Y ) is compatible with S(X) = 1 over G. We say
that ©(X; Y ) admits a lift to a generic point of S = 1 over G (or shortly S-lift over G)
if 9Y ©(X ¹ ; Y ) is true in GR(S) (here Y are variables and X ¹ are constants from GR(S) ).
The next result tells one that lifting of atomic formulas is equivalent to lifting of some
equations.
Proposition 5 Let G be a non-Abelian group discriminated by free groups. Suppose
that S(X) = 1 be an equation over G and
©(X; Y ) =
n
_
i=1
(Ti (X; Y ) = 1 & Wi (X; Y ) 6
= 1):
25
be an atomic formula over G which is compatible with S(X) = 1. Then ©(X; Y ) admits
W
S-lift if and only if the formula ni=1 (Ti (X; Y ) = 1 admits S-lift which is equivalent to
lifting a particular equation T (X; Y ) = 1 over G (which can be e®ectively constructed
from ©(X; Y )):
Proof. If 9Y ©(X ¹ ; Y ) is true in GR(S) then obviously the formula
¹
T (X ; Y ) =
n
_
(Ti (X ¹ ; Y ) = 1)
i=1
is true in GR(S) for the same value of Y , i.e., T (X; Y ) admits S-lift.
Suppose now that the formula T (X; Y ) admits S-lift. If T1 (X; Y ) = 1 does not admit
S-lift then 8Y (T (X ¹ ; Y ) 6
= 1) in GR(S): It implies, that the formula
ª(X ¹ ; Y ) =
n
_
i=2
(Ti (X ¹ ; Y ) = 1 & Wi (X ¹ ; Y ) 6
= 1):
holds in GR(S) , so ª(X; Y ) admits S-lift and we argue by induction on the number of
disjunctions in ª(X; Y ).
Suppose now that the equation T1 (X; Y ) = 1 admits S-lift (this case includes the
base of induction). If the formula
9Y (T1 (X ¹ ; Y ) = 1 & W1 (X ¹ ; Y ) 6
= 1)
(12)
holds in GR(S) then the formula ©(X ¹ ; Y ) holds in GR(S) , i.e., it admits S-lift. Assume,
that the formula (12) does not hold in GR(S) . It implies, that the formula
8Y (T1 (X ¹ ; Y ) = 1 ¡! W1(X ¹ ; Y ) = 1)
(13)
holds in GR(S) . Hence, W1 (X ¹ ; Y ) belongs to the radical Rad(T1 ) over the group GR(S):
By Proposition 4 the group GR(S;T1 ) is the coordinate group of T1(X ¹ ; Y ) over GR(S),
hence for any homomorphism
Á : GR(S;T1 ) ¡! G
the image W1 (X; Y )Á is trivial. It implies, that for any tuples a¹; ¹b over G if S(¹a) = 1
and T1 (¹a; ¹b) = 1 then W1 (¹a; ¹b) = 1. If n = 1, then this contradicts to the compatibility
of S(X) = 1 and T1 (X; Y ) = 1 & W1 (X; Y ) 6
= 1 - hence the base for induction holds. If
n > 1 then it follows that the formula ª(X; Y ) is equivalent to ©(X; Y ) over G and we
again can proceed by induction. 2
26
6
Implicit function theorems
6.1
Lifting equations into a generic point of a standard quadratic
equation over a freely discriminated group
Let F = Fm (A) be a free group of rank m with basis A:
In this subsection we prove the following theorem.
Theorem 2 Let S(X; A) = 1 be a standard quadratic equation over the group G with
the set of variables X, such that either S = [x; y]d¡1, or S = [x; y][z; t] or the equation
¡1
S = 1 has a non commutative solution (see de¯nition 13 in [15]) and S 6
= cz11 cz22 c¡1
2 c1 ;
¡2
S6
= x2cz c¡1a¡2 ; S 6
= x21x22 a¡2
2 a1 : Then every equation T (X; Y; A) = 1 compatible with
S(X; A) = 1 admits S-lift.
Proof. We divide the proof into two parts: the ¯rst one for orientable quadratic equation
S = 1 and the second one for non-orientable.
Part I. Let S = 1 be a standard orientable quadratic equation
S=
m
Y
zi¡1 ci zi
i=1
n
Y
[xi ; yi ]d¡1 = 1
i=1
(here ci ; d are constants from G), satisfying the condition of the theorem. If xi = ai ; yi =
Q
Qn
¡1
bi ; zi = ei is a solution of this equation in G, then d = m
i=1 ei ci ei
i=1 [ai ; bi ]: Hence we
can write the equation in the following form:
m
Y
i=1
zi¡1ci zi
n
Y
[xi ; yi ] =
i=1
m
Y
e¡1
i ci ei
i=1
n
Y
[ai ; bi ]:
(14)
i=1
Moreover, we will always assume (except for the equations [x; y] = d and [x1; y1 ][x2; y2 ] =
1) that the solution ai ; bi ; ei is in a general position (see [15] for de¯nitions).
Recall (see introduction above) that if H is a group, u 2 H, and CH (u) is the
centralizer of u in H, then by H(u; t) we denote the following HNN-extension
H(u; t) =< H; t j t¡1 xt = x (x 2 CH (u)) >;
which is called an extension of the centralizer of u. If
G = G1 · G1 (u1; t1) = G2 · : : : · Gn (un ; tn ) = Gn+1
27
is a chain of extensions of centralizers of elements ui 2 Gi , then we denote the resulting
group Gn+1 by G(U; T ), where U = fu1 ; : : : ; un g and T = ft1; : : : ; tn g.
Now, by induction, we de¯ne a sequence of consequent extensions of centralizers and
a sequence of group homomorphisms in the following way.
Suppose that m 6
= 0: In this event for each i = 1; : : : ; 2m ¡ 1 we de¯ne by induction a
pair (µi ; Hi ) , consisting of a group Hi and a homomorphism µi from GS into Hi . These
homomorphisms µi 's transform only z-variables and leave x's and y's ¯xed. Before we
will go into formalities let us explain the idea that lies behind this. If (e1; : : : ; em ) is a
solution of the equation
¡1
z1¡1c1 z1 : : : zm
cm zm = c1 : : : cm ;
(15)
then transformations
ei ! ei (cei i )p ; ej ! ej
ei !
ei+1 q
ei (cei i ci+1
);
ei+1 !
ei+1 q
ei+1 (cei i ci+1
);
(j 6
= i);
ej ! ej
(16)
(j 6
= i; i + 1);
(17)
also give a solution of the equation (15) for arbitrary integers p; q. To avoid collapses
under cancelation one might want to have numbers p; q as big as possible, the best way
would be to have p; q = 1. Since there are no in¯nite powers in G, to realize this idea
one should go outside the group G and represent the in¯nite power of an element, say u,
by a new letter t which commutes with u. That is where the extensions of centralizers
came from. Now we give formal de¯nitions for the G- homomorphisms µi 's and the
groups Hi 's.
For i = 0 put
H0 = G; xi µ0 = ai ; yi µ0 = bi ; zi µ0 = ei ;
(if n = 0 then xi 's and yi 's do not occur in S = 1 and we simply omit them in the
de¯nition above).
Suppose now, that groups H2i and G-homomorphisms µ2i are already de¯ned. We
are going to perform ¯rst the transformation (16) and then the transformation (17)
(more precisely, their analogs in the corresponding extensions of centralizers). Namely,
we de¯ne
µ2i
zi+1
H2i+1 =< H2i ; r2i+1 j[CH2i (ci+1
); r2i+1 ] = 1 >;
and
µ
µ2i
2i+1
zi+1
= zi+1
r2i+1 ;
28
sµ2i+1 = sµ2i
where s is an arbitrary variable in S = 1 which is not equal to zi+1 . Furthermore,
µ
H2i+2 =<
µ
µ
2i+1
2i+1
zi+1
zi+2
H2i+1; r2i+2 j[CH2i+1 (ci+1
ci+2
); r2i+2 ]
µ
µ
µ
2i+2
2i+1
2i+2
2i+1
zi+1
= zi+1
r2i+2; zi+2
= zi+2
r2i+2 ; sµ2i+2 = sµ2i+1 ;
= 1 >;
(s 6
= zi+1 ; zi+2 ):
Observe, that by the de¯nition above the element r2i+1 extends the centralizer of the
ei+1 r2i
element ci+1
from the group H2i and the element r2i+2 ; extends the centralizer of the
ei+1 r2i r2i+1
element ci+1
ci+2 from the group H2i+1 : In particular the following equalities hold
in the group H2m¡1 for each i = 0; : : : ; 2m ¡ 1:
ei+1 r2i
ei+1 r2i r2i+1
[r2i+1 ; ci+1
] = 1; [r2i+2; ci+1
ci+2 ] = 1:
(18)
(for convenience we de¯ne here r0 = 1). Observe also, that the resulting mapping µ2m¡1
acts on z1 ; : : : ; zm 2 GS as follows
z1 ! e1 r1r2 ; zi ! ei r2(i¡1) r2i¡1 r2i ; zm ! em r2(m¡1) r2m¡1
(0 < i < m):
(19)
[ai ; bi ];
(20)
From equations 18 and 19 it readily follows that
(
m
Y
zi¡1ci zi
i=1
n
Y
¡1
[xi ; yi ])µ2m¡1 = e¡1
1 c1 e1 : : : em cm em
n
Y
i=1
i=1
and the map µ2m¡1 gives rise to a G-homomorphism (which we again denote by µ2m¡1 )
µ2m¡1 : GS ¡! H2m¡1:
The group H2m¡1 is obtained from the initial group G by ¯nitely many extensions of
centralizers. If n = 0 we denote H2m¡1 by G(U; T ), according to our notations for chains
of extensions of centralizers, where U = fu1 ; : : : ; u2m¡1 g, T = fr1; : : : ; r2m¡1 g and
u1 = c1 ; u2 = c1 c2; u3 = cr22 ; u4 = cr22 c3 ; u5 = cr34 ; : : : ;
e
r
e
i+1 2i
i+1
u2i+1 = ci+1
; u2i+2 = ci+1
r2i r2i+1
em r2(m¡1)
ci+2 ; : : : ; u2m¡1 = cm
:
Also we denote G homomorphism µ2m¡1 by ¸. Notice, that we constructed ¸ starting
from an arbitrary solution µ0 of the equation S = 1 provided this solution is in a general
position. Sometimes we will write ¸¯ emphasizing the fact that we constructed ¸ starting
on the given solution ¯.
29
Suppose now, that n > 0, i.e. in the equation S = 1 we have non-trivial commutator part [x1 ; y1 ] : : : [xn ; yn ]. Similar to the case above we start with the principal
transformations. They consist of two Dehn's twists:
x ! yp x; y ! y;
(21)
x ! x; y ! xpy;
(22)
which leave the commutator [x; y] unchanged, and the third transformation which ties
two consequent commutators [xi ; yi ][xi+1 ; yi+1 ]:
¡q
¡1 ¡q
¡1 q
xi ! (yi x¡1
i+1 ) xi ; yi ! (yi xi+1 ) yi (yi xi+1 ) ;
(23)
¡q
¡1 q
¡1 ¡q
xi+1 ! (yi x¡1
i+1 ) xi+1 (yi xi+1 ) ; yi+1 ! (yi xi+1 ) yi+1 :
Now we de¯ne by induction on i for i = 0; : : : ; 4n ¡ 1 pairs (Gi ; ®i ) consisting of
groups Gi 's and G-homomorphisms ®i 's from GS into Gi .
First we have to connect the commutator part of the equation S = 1 with the product
of conjugates. For this sake de¯ne
µ2m¡1
G0 =< H2m¡1 ; r2m j[CH2m¡1 (czmm
2m¡1
x¡µ
); r2m ] = 1 >;
1
®0
µ2m¡1
¡1
= zm ; x1 ; y1 ):
zm
= zm
r2m ; x®1 0 = a1r2m ; y1®0 = r2m
b1 ; s®0 = sµ2m¡1 (s 6
After that for each commutator [xi ; yi ] in S = 1 we will perform consequently three
Dehn's twists ( 22), ( 21), ( 22) (more precisely, their analogs for an extension of a
centralizer) and an analog of the connecting transformation ( 23) provided the next
commutator exists. Namely, suppose G4i and ®4i have been already de¯ned.
Then
4i
G4i+1 =< G4i ; t4i+1 j[CG4i (x®i+1
); t4i+1 ] = 1 >;
®4i+1
®4i
= yi+1 ):
yi+1
= t4i+1 yi+1
; s®4i+1 = s®4i (s 6
®
4i+1
G4i+2 =< G4i+1; t4i+2 j[CG4i+1 (yi+1
); t4i+2 ] = 1 >;
®
®
4i+2
4i+1
xi+1
= t4i+2 xi+1
; s®4i+2 = s®4i+1
(s 6
= xi+1 ;
4i+2
G4i+3 =< G4i+2; t4i+3 j[CG4i+2 (x®i+1
); t4i+3 ] = 1 >;
30
®
®
4i+3
4i+2
yi+1
= t4i+3yi+1
; s®4i+3 = s®4i+2
®
(s 6
= yi+1 );
¡®
4i+3
G4i+4 =< G4i+3 ; t4i+4j[CG4i+3 (yi+1
xi+24i+3 ); t4i+4] = 1 >;
®4i+3
®4i+4
®4i+3 t4i+4
®4i+3 t4i+4
4i+4
4i+4
x®i+1
= t¡1
= yi+1
; x®i+2
= xi+2
;
4i+4 xi+1 ; yi+1
®
®
4i+4
4i+3
®4i+4
yi+2
= t¡1
= s®4i+3 (s 6
= xi+1 ; yi+1 ; xi+2; yi+2 ):
4i+4 yi+2 ; s
Thus we have de¯ned groups Gi and mappings ®i for all i = 0; : : : 4n¡1 in the case when
m6
= 0 and n 6
= 0. As above, the straightforward veri¯cation shows that the mapping
®4n¡1 gives rise to a G-homomorphism ®4n¡1 : GS ¡! G4n¡1 : Put G(U; T ) = G4n¡1 and
denote ®4n¡1 by ¸. Thus we have a G-homomorphism
¸ : GS ¡! G(U; T ):
Suppose now, that m = 0 and n 6
= 0. In this case de¯ne
®0 : xi ! ai ; yi ! bi ;
i.e., ®0 is an arbitrary solution of the equation S = 1 in a general position. After that we
de¯ne Gi ; ®i and G(U; T ); ¸ as above. Again, as before, the homomorphism ¸ depends on
the choice of the initial solution. Writing ¸¯ emphasizes that fact that ¸ is constructed
starting from an initial solution ¯.
Now we are ready to formulate the following
Proposition 6 For an arbitrary solution in a general position ¯ of the equation
S = 1 the homomorphism ¸¯ : GS ! G(U; T ) is an G-embedding.
Proof. In the proof of this proposition we will use induction on the atomic rank of
the equation in the same way as in the proof of Theorem 1 in [15]. We have to consider
G as an arbitrary fully residually free group and for equation (14) consider the following
cases for S:
1. n = 1; m = 0; prove that à = ®3 is an embedding of Gs into G3
2. n = 2; m = 0; prove that à = ®4 is a monomorphism on H =< G; x1 ; y1 > :
3. n = 1; m = 1; prove that à = ®0 is a monomorphism on H =< G; z1 > :
4. n = 0; m = 2; prove that µ3 is an embedding of GS into H2 (this case is considered
in [15], Proposition 13).
31
Case 1. Choose an arbitrary nontrivial element h 2 Gs . It can be written in the
form
h = g1 v1 (x1 ; y1) g2 v2 (x1 ; y1 ) g3 : : : vn (x1 ; y1 ) gn+1 ;
where 1 6
= vi (x1 ; y1 ) 2 F (x1 ; y1) are words in x1; y1 , not belonging to the subgroup
< [x1; y1 >]; and 1 6
= gi 2 G; gi 6
2< [a; b] > (with the exception of g1 and gn+1, they
could be trivial). Then
hà = g1 v1 (t3 t1a; t2 b) g2 v2 (t3 t1a; t2 b) g3 : : : vn (t3t1 a; t2b) gn+1:
(24)
The group G(U; T ) is obtained from G by three HNN-extensions (extensions of centralizers), so every element in G(U; T ) can be rewritten to its reduced form by making ¯nitely
many pinches.
It is easy to see that the leftmost occurrence of either t3 or t1 in the product (24)
occurs in the reduced form of hà uncanceled.
Case 2.
¡1
¡1
¡1
x1 ! t¡1
4 t2 a1 ; y1 ! t4 t3 t1 b1 t4 ; x2 ! t4 a2 t4 ; y2 ! t4 b2 :
Choose an arbitrary nontrivial element h 2 H = G ¤ F (x1 ; y1 ). It can be written in
the form
h = g1 v1 (x1 ; y1) g2 v2 (x1 ; y1 ) g3 : : : vn (x1 ; y1 ) gn+1 ;
where 1 6
= vi (x1 ; y1 ) 2 F (x1 ; y1 ) are words in x1 ; y1, and 1 6
= gi 2 G: (with the exception
of g1 and gn+1 , they could be trivial). Then
¡1
t4
¡1
t4
t4
hà = g1 v1 (t¡1
4 t2 a; (t3 t1 b) ) g2 v2 (t4 t2 a; (t3 t1 b) ) g3 : : : vn (t4 t2 a; (t3 t1 b) ) gn+1 : (25)
The group G(U; T ) is obtained from G by four HNN-extensions (extensions of centralizers), so every element in G(U; T ) can be rewritten to its reduced form by making ¯nitely
many pinches. It is easy to see that the leftmost occurrence of either t4 or t1 in the
product (25) occurs in the reduced form of hà uncanceled.
Case 3.
We have an equation cz [x; y] = c[a; b],
z ! zr1r2 ; x ! ar2 ; y ! r2¡1b: and [r1 ; c] = 1; [r2; ca¡1 ] = 1: Here we can always
suppose, that [c; a] 6
= 1, by changing a solution, hence [r1 ; r2] 6
= 1:
The proof for this case is a repetition of the proof of Proposition 11 in [15]. 2
De¯nition 8 Let G(U; T ) be the group constructed just before Proposition 1. Then
if we denote by u1 ; : : : ; u2m ; v1; : : : ; v4n¡1 the elements, centralizers of which we extend,
we have
G(U; T ) = G(r1 ; : : : ; r2m ; t1; : : : ; t4n¡1; u1 = c1 ; u2 = c1 c2; u3 = cr22 ; u4 = cr22 c3 ;
32
r2m
¡1
u5 = cr34 ; : : : ; u2m = crm2m¡1 a¡1
1 ; v1 = a1 ; : : : ; v4n¡1 = t4n¡2 t4n¡4 bn t4n¡4 ):
i
Denote by Ã(M1 ;:::;M2m ;M2m+1 ;:::;M2m+4n¡1 ) a G-homomorphism sending ri in uM
and
i
M2m+j
tj into vj
: Denote by Ã(M1 ;:::;Mk ) ; where k = s(2m + 4n ¡ 1), a composition of s
homomorphisms Ã(Mi(2m+4n¡1)+1 ;:::;M(2m+4n¡1)(i+1) ) , where i = 1; : : : s:
A homomorphism Ã(M1 ;:::;Mk ) ; is called standard.
A family of standard G-homomorphisms is called positive unbounded, if all Mi are
positive and for any number N there is a homomorphism Ã(M1 ;:::;Mk ) in this family such
that each Mi is greater than N.
A standard G-homomorphism is called positive N -large, if all Mi are positive and
greater than N.
A family of solutions is called positive unbounded, if corresponding homomorphisms
form a positive unbounded family.
A solution is called positive N -large if the corresponding homomorphism is N-large.
Corollary 8 A restriction on GS of any positive unbounded family of standard Ghomomorphisms from G(U; T ) onto G is a discriminating family.
Suppose now that the equation has form (14). We will prove Theorem 2 ¯rst for the
special case n = 1; m = 0. Remind that we constructed à = ®4n¡1 ±: : :±®0 ±µ2m¡1 ±: : :±µ1:
The following lemma has place.
Lemma 10 Let ai ; bi ; ci be a solution of the equation (14). It is possible to ¯nd large
positive ki ; mi ; ni << ki+1 ; mi+1; ni+1 , with the property that if instead of this solution we
m
m
take a^i = (bni i ai )([ai ;bi ]) i ; ^bi = ((bni i ai )ki bi )([ai ;bi ]) i then when we make all possible can^ ^ a¡1
celations in words a^i^bi ^ai , a^¡1
^i+1 ^ai , a^¡1
^i+1^bi+1 , ^bi a^¡1
^i , ^bi¡2 ^a¡1
^i ; ^bi a^i^b¡1
i a
i a
i+1 a
i¡1 bi ; bi¡2 ^
i¡1 a
i¡1
none of the factors will be completely canceled.
We can also change the solution so that in the products ci ci¡1 and ci c¡1
i¡1 none of the
factors is completely canceled.
Follows from the fact that ai ; bi ; ci is a solution in the general position.
Let S(X; A) = 1 be a standard quadratic equation of type (14). Consider ¯rst the
case, when n = 1; m = 0. The equation has form
[x; y] = [a; b];
where [a; b] 6
= 1:
33
Consider now all the solutions of this equation in G = F obtained from the initial
one by application of the initial segment of à k for large k's and taking standard Ghomomorphisms. Denote the result xÁi ; y Ái :
Then xÁ0 = a; yÁ0 = b;
xÁ2i+1 = xÁ2i ; yÁ2i+1 = (xÁ2i )n2i+1 y Á2i ;
xÁ2i+2 = (y Á2i+1 )n2i+2 xÁ2i+1 ; y Á2i+2 = y Á2i+1 :
For example, xÁ5 = (((an1 b)n2 a)n3 an1 b)n4 (an1 b)n2 a;
y Á5 = ((((an1 b)n2 a)n3 an1 b)n4 (an1 b)n2 a)n5 ((an1 b)n2 a)n3 an1 b:
By Theorem 10 there exists a ¯nite number of cuttings for the variables from X for
the system of equations [rj=1 vj (X; Y; A) = 1&S(X; A) = 1 such that each X which is a
projection of a solution X; Y of this system, must satisfy one of these cuttings.
We can therefore suppose that the interval I for each cutting consists of the union of
the intervals corresponding to z Ái , xÁi ; y Ái : Each variable is partitioned in several ways
into variables corresponding to bases ¸i 's.
The same generalized equation will serve for some positive unbounded family of
solutions.
If some base ¸i occurs only once, then we can disregard this partition, because the
other bases in this cutting can be found from other partitions, and ¸i can be expressed
in terms of them.
Let this generalized equation - contain N cutting points (if some boundary of - is
a cutting point, then it is counted as a cutting point only once). Consider some solution
Ák belonging to the positive unbounded family, satisfying - , such that k > N . This
solution has an obvious periodic structure.
Let for k = 2i Ak = yÁ2 i¡1 and for k = 2i + 1 Ak = xÁ2 i : We call Ak the oldest period
in rank k. The rank of a cutting is determined by the rank of the oldest period. Ak is
not necessary cyclically reduced, denote by A0k cyclically minimal word conjugated to
Ak . We say that a variable in rank k has a long value if it contains A0k , otherwise it has
a short value. Abusing the language we will call them long and short variables in rank
k.
Consider for de¯niteness the cutting of the equation in rank M = 2i + 1. Then
ÁM
x = AM ; y ÁM = AnMM y ÁM¡1 :
If AM = c¡1A0M c, where A0M is cyclically reduced we consider for yÁM the interval,
where the graphical product (c¡1 A0M c) ± : : : ± (c¡1 A0M c) ± (c¡1A0M cyÁM ¡1 ) is written (the
word written on this interval is not reduced anymore, but we can only cancel c¡1 c).
De¯nition 9 A cutting point on such an interval is called an integral point, if it
corresponds to the beginning or to the end of AM .
34
We can replace the cutting of - by a system §s;i of linear equations in integral
variables (variables are powers of AM ) and by a cutting in the rank M ¡ 1 . The new
interval IM ¡1 will be the union of subintervals JM j which are obtained by cutting out
from IM all the occurrences of AM which do not contain any cutting points. These
Á
Á
subintervals correspond to the following elements: xÁi M = xi M ¡1 ; yi M ¡1 ; and a ¯nite
sÁ
sÁ
Á
number of words zi M ¡1 ; xi M ¡1 yi M¡1 (s · N).
Obviously, the number of cutting points in rank M ¡1 is not greater than the number
of points in rank M . If one of the cutting points is an integral point in rank M , then
the number of cutting points in rank M ¡ 1 would decrease.
Consider the parameter K = (k1 ; k2 ; : : : ; kn ; : : : ; kN ); where kn is the number of
subintervals with n cutting points. The tuples are ordered lexicographically from the
right to the left.
Suppose that after 4 steps, when we go from rank K1 to rank K1 ¡4 parameter K does
not decrease (in particular, there is no integral cutting points in these ranks). Denote
K2 = K1 ¡ 3. Suppose, for de¯niteness, that K2 is even. Denote ¹a = xÁK2 ; ¹b = yÁK2 :
Then
xÁK2 +1 = a¹; yÁK2 +1 = a¹nK2 +1 ¹b;
xÁK2 +2 = (¹
anK2 +1 ¹b)nK2 +2 a¹; yÁK2 +2 = a¹nK2 +1 ¹b;
xÁK1 = (¹
anK2 +1 ¹b)nK2 +2 ¹a;
ÁK1
y
= ((¹
anK2 +1 ¹b)nK2 +2 a¹)nK1 a¹nK2 +1 ¹b:
Consider now variables corresponding to the cutting in rank K1 as original variables.
The subintervals in rank K1 will correspond only to xÁK1 and y ÁK1 , otherwise K
would decrease on the next step. The same can be said about all ranks from K1 to K2.
This means that all the cutting points must be inside ¹a and ¹b. We call the variables
short in rank K2 just short variables. Then all the equations of the cutting in rank
K1 have form y ÁK1 = d1 d2 : : : dr or xÁK1 = d01 d02 : : : d0t ; where d1 ; d01 ; dr ; d0t may be long
variables, and all other variables are short. Notice, that xÁK1 and y ÁK1 can begin with
the same long variable, but cannot end on the same long variable. If they begin with
the same long variable, then y ÁK1 must end on a long variable which does not appear in
any cutting of xÁK1 : If some equation begins with a long variable d1 then no equations
end on d1 . xÁK1 cannot begin and end with a long variable.
If yÁK1 = d1 d2 : : : dr or xÁK1 = d1 d2 : : : dr ; then we call d1 and dr the side variables.
Consider the equivalence relation R on the set of all left side variables and right side
variables generated by the following: (d1; d01 ) 2 R if there exists a partition of J = y ÁK1
or J = xÁK1 such that J = d1 d2 : : : dr¡1dr = d01d02 : : : d0p¡1 dr ; (dr ; d0p ) 2 R if there exists a
partition of J such that J = d1 d2 : : : dr¡1 dr = d1 d02 : : : d0p¡1 d0p :
35
Fix one representative in each equivalence class. Suppose it is a class of long variables
that occur on the right, and dr is the representative. Then each other variable d0p in this
class is the product ddr , where d is the product of some short variables. We also will
have some equations among short variables.
The set of all partitions d1 : : : dr corresponding to the same interval J can be also
divided into equivalence classes: two such partitions are equivalent if the right (and left)
side variables are equivalent. Every two distinct partitions in the same equivalence class
imply an equation on the short variables.
It is enough to express only one long variable in each equivalence class in terms of
the short variables and yÁK1 and xÁK1 .
We can consider the graph ¡ having as vertices the equivalence classes of partitions of
y ÁK1 and xÁK1 and having two vertices connected by an edge if they have the equivalent
right or left long side variables. This graph contains maximum one edge, connecting
possibly the two equivalence classes including partitions of y ÁK1 and xÁK1 with the same
left long variables.
First we express the long variables from the partitions of xÁK1 in terms of xÁK1 and
short variables. Then substitute the left long variable corresponding to the edge of the
graph into the partitions of y ÁK1 and express the other long variables in terms of this
one, y ÁK1 , and short variables.
To summarize: if we solve in F the system of equations in short variables, then the
long variables can be expressed in terms of xÁK1 ; y ÁK1 and short variables. The family
of solutions xÁK1 ; yÁK1 is a discriminating family, hence the same formula will work
for each solution x; y of the equation [x; y] = [a; b]: Hence the variables in rank K1 are
expressed in terms of constants (short variables) and x and y. The original variables
can be expressed in terms of the variables in rank K1 and x; y:
Consider the general case now. We need the following de¯nition. Let
Ãs = Ã(Ms;1 ;:::;Ms;2m ;Ms;2m+1 ;:::;Ms;2m+4n¡1 ) :
Denote by ¯0 the solution xi = ai ; yi = bi ; ci = 1 of S = 1, Let ¯i be a solution given
by
¯i = Ãi ± ¸¯i¡1 :
Let M = (2m + 4n ¡ 1)(s ¡ 1) + p. Denote by ÁM the solution ¯s such that in Ãs
the numbers Ms;p+1 ; : : : ; Ms;2m+4n¡1 are zeros.
De¯nition 10 If we have a solution ÁM ; where Ms;i is the last nonzero number, and
M = (2m + 4n ¡ 1)(s ¡ 1) + i, then the oldest period of rank M is a specialization of the
generator of the centralizer of ts;i , if i > 2m and rs;i if i · 2m.
36
A rank of the partition of a word in variables zjÁs ; xÁi s ; yiÁs ; j = 1; : : : ; m; i = 1; : : : ; n
is the rank of the oldest period of this partition.
We will take each Ms;2i+1 = 1 for s > 1; i = 0; : : : ; m ¡ 1 equal 0. This way we will
have the same family of solutions as in general case. Then Á2m+4n¡1+2i = Á2m+4n¡1+2i+1
for i = 0; : : : ; m ¡ 1:
Let K1 = (2m + 4n ¡ 1)(r + 1) and K2 = (2m + 4n ¡ 1)r.
Á
Á
Á
Denote z¹i = zi K2 ; ¹ai = xi K2 ; ¹bi = yÁK2 : Then, if we denote z~i = zi K2 +2i¡1 , we have
Á
z1 K1 = z¹1 (cz1¹1 cz2¹2 )m1 ;
:::::
Ã
!mi
z
~
z
¹
i¡1 i mi¡1
z¹i (ci¡1
ci )
z¹i+1
c¹i+1
Á
~i¡1 z¹i mi¡1
ci
zi K1 = z¹i (czi¡1
ci )
for i = 2; : : : ; m ¡ 1:
We have also
Á
~i¡1 z¹i mi¡1
zi K1 = z¹i (czi¡1
ci )
for i = m; n = 0; and Ã
Á
zi K 1
=
z
~
z
¹
z¹i (c i¡1 c i )mi¡1 ¡1
ci i¡1 i
a¹1
~i¡1 z¹i mi¡1
z¹i (czi¡1
ci )
for i = m; n 6
= 0:
For x1 ; y1 weà have:
z
~
m
z
¹
z
¹n (c m¡1 cmm ) m¡1
a
¹¡1
cm m¡1
1
Á
x1 K2 +2m = a¹1Ã
Á
Á
Á
;
!n10
z
~m¡1 z
¹m mm¡1
¡¹
zm (cm¡1
cm
)
y1 K2 +2m+1 = a¹1 cm
!n10
;
!n;10
Á
¹n1 11 ¹b1 ;
a
Á
K2 +2m+1
1
x1 KÃÃ
= x1 K2 +2m+2 = (y1 K2 +2m+1
)n12 x1 !
!
=
Ã
¹an1 11 ¹b1
¹m mm¡1
¡¹
zm (c m¡1 czm
)
a¹1cm m¡1
!n10
Á
Á
z
~
Á
n12
n10
z
~
¹m mm¡1
¡¹
zm (c m¡1 czm
)
¹a1 cm m¡1
a¹1
Ã
z
~
¹m mm¡1
z¹m (c m¡1 czm
)
cm m¡1
a¹¡1
1
!n10
;
Á
2 +2m+2 n12
y1 KÃ1 = y1 K2 +2m+3 = (x1 K!
) y1 K2 +2m+1
z
~
¹m mm¡1
m¡1 z
¡¹
zm (cm¡1
cm
)
= ¹a1 cm
n10
a¹n1 11 ¹b1
0ÃÃ
1n13
!n10
!n12 ¡1 Ã
!n10
z
~m¡1 z
z
~m¡1 z
¹m mm¡1
¹m mm¡1
¡¹
zm (cm¡1
cm
)
¡¹
z
(c
c
)
m
m
@
a¹1 cm
a¹n1 11 ¹b1
¹a1 cm m¡1
¹n1 11 +1¹b1 A :
a
Á
K2 +2m+4i¡5
And ¯nally if we denote by ~bi¡1 = yi¡1
; we have
37
Á
n0 ¹
yi K2 +2m+4i¡4 = (¹
ai~b¡1
i¡1 ) bi ;
Á
n0
n0
xi K2 +2m+4i¡4 = (¹
ai~b¡1
ai (~bi¡1a¹¡1
i¡1 ) ¹
i ) ;
Á
n0 n1 ¹
yi K2 +2m+4i¡3 = (¹
ai~b¡1
ai bi ;
i¡1 ) ¹
Á
Á
n0 n1 ¹ n2
n0
n0
xi K2 +2m+4i¡2 = xi K2 +2m+4i¡1 = ((¹
ai~b¡1
ai bi ) (¹
ai~b¡1
ai (~bi¡1a¹¡1
i¡1 ) ¹
i¡1 ) ¹
i ) ;
Á
n0 n1 ¹
n0 n1 ¹ n2 ¡1
n0 n1 +1¹ n3
¹i bi (((¹
ai bi )
¹i bi ) ;
yi K2 +2m+4i¡1 = (¹
ai~b¡1
ai~b¡1
(¹ai~b¡1
i¡1 ) a
i¡1 ) ¹
i¡1 ) a
Á
¡ÁK2 +2m+4i¡1 n4 ÁK2 +2m+4i¡1
;
) xi
Á
xi K1 = xi K2 +2m+4i = (¹
ai+1yi
Á
Á
¡ÁK2 +2m+4i¡1 n4 ÁK2 +2m+4i¡1 ÁK2 +2m+4i¡1 ¡1 n4
) yi
(yi
¹ai+1 ) :
yi K1 = yi K2 +2m+4i = (¹ai+1yi
Denote by As the oldest period in rank s. Then, for example,
n0 n1 ¹
n0 n1 ¹
n0 n1 +1¹ n3 ¡1
AK2 +2m+4i = ((¹ai~b¡1
¹i bi )(((¹
ai~b¡1
ai bi ))n2 ¡1 (¹ai~b¡1
ai bi ) a¹i+1 .
i¡1 ) a
i¡1 ) ¹
i¡1 ) ¹
0
As is not necessary cyclically reduced, denote by As cyclically minimal word conjugated to As . As above, a variable is long in rank s if it contains A0s . Otherwise it is
called short.
By Lemma 10 we can suppose that when we make all possible cancelations in words
¹
¹ ¹ a¡1
a¹i bi a¹i , ¹a¡1
¹i+1 a¹i , a¹¡1
¹i+1¹bi+1 , ¹bi ¹a¡1
ai , ¹bi¡2 ¹a¡1
ai ; ¹bi a¹i¹b¡1
¹i c¹i¡1 c¹i ; c¹i c¹¡1
¹i
i a
i a
i+1 ¹
i¡1 bi ; bi¡2 ¹
i¡1 ¹
i¡1 ; c
i¡1 c
none of the factors will be completely canceled.
Consider the cutting of equation in rank M.
Every variable J = xÁi M ; yiÁM ; ziÁM can be uniquely represented as J = ApM v or
J = uApM or J = ApM1 wApM2 , where p (p1 ; p2 ) is maximal possible such power, u; v; w do
not contain powers of AM .
It can happen that AM = c¡1 A0M c, in this case, instead of the interval corresponding
to the variable J we consider the interval, where the graphical product AM ± : : : ± AM ±
(AM v) or (uAM ) ± AM ± : : : ± AM or AM ± : : : ± (AM wAM ) ± : : : ± AM is written (the word
written on this interval then is not reduced anymore, but we can only cancel cc¡1 ).
As above, we can replace the cutting of - by a system §s;i of linear equations in
integral variables (coe±cients p's of powers of A) and by a cutting in the rank M ¡ 1
(u's and v's will be new bases). The new interval IM ¡1 will be the union of subintervals
38
JM j which are obtained by cutting out from IM all the occurrences of AM which do not
contain any cutting points. These subintervals correspond to the following elements:
Á
Á
Á
Á
Á
Á
xi M¡1 ; yi M ¡1 ; zi M¡1 and a ¯nite number of words in xi M ¡1 ; yi M¡1 ; zi M¡1 . These words
Á
Á
can be AM itself, ¯nite powers As (s · N) and words of the form xi M ¡1 As , yi M ¡1 As ,
Á
Á
Á
Á
zi M¡1 As , As xi M ¡1 , As yi M ¡1 , As zi M ¡1 , where s < N .
The number of cutting points in rank M ¡ 1 is not greater than the number of points
in rank M .
Consider the parameter K = (k1; k2; : : : ; kn ; : : : ; kN ) de¯ned above.
Lemma 11 Suppose that when we go from rank
K1 = (2m + 4n ¡ 1)(r + 1) to K1 ¡ 2(2m + 4n ¡ 1) parameter K is not decreasing.
Denote K2 = (4m + 2n ¡ 1)r.
Then the variables corresponding to the bases of the cutting in rank K1 can be divided
into two families: short variables Z (in rank K2) and long variables. The long variables
Á
Á
Á
can be expressed in terms of xi K1 ; yi K1 ; ci K1 and Z. Variables Z are solutions of some
system of equations.
Á
Á
Á
De¯nition 11 If the numbers K1 and K2 are as above, then xi K1 ; yi K1 ; zi K1 are
called the core part of xÁi M ; yiÁM ; ziÁM
Proof. Suppose that K does not decrease in ranks K1 ; : : : ; K1 ¡ 2(2m + 4n ¡ 1). Now
we consider as original variables the variables corresponding to rank K1 .
Let s; s1 2 fK2 ; : : : ; K1 g; s < s1 . The following facts are important for our proof.
Fact 1.
Every subinterval J in rank s can contain not more than two long variables ,because
otherwise K is decreasing. Also J should begin or end with a long variable.
Fact 2.
Suppose we have a partition J = d1 d2 : : : dr¡1 dr , in rank s1. Then this partition
will induce in rank s the following partition J 0 = d01 d2 : : : dr¡1d0r . If d1 is short in ranks
s + 1; : : : ; s1 , then d1 = d01 . If d1 is long in some rank s1 > s; then d01 does not appear in
the intervals of rank s that are not induced by the cuttings of ranks s + 1; : : : ; s1.
Fact 3.
Suppose we have in rank s: J = d1 d2 : : : dr , and J = d1 g2 : : : gs . Then d2; : : : ; dr¡1
and g2; : : : ; gs¡1 are short in all ranks greater than K2 and hence there is a product of
short variables d, such that dr = dgs .
If J = d1 d2 : : : dr , then we call d1 and dr the side variables and corresponding bases
the side bases.
39
Fact 4. Consider the equivalence relation R on the set of all left side variables and
right side variables in rank k generated by the following: (d1 ; d01 ) 2 R if there exists a
partition of J of rank k such that J = d1 d2 : : : dr¡1 dr = d01 d02 : : : d0p¡1 dr ; (dr ; d0p) 2 R if
there exists a partition of J of rank k such that J = d1 d2 : : : dr¡1 dr = d1 d02 : : : d0p¡1d0p:
Fix one representative in each equivalence class. Suppose it is a class of long variables
that occur on the right, and dr is the representative. Then each other variable d0p in this
class is the product ddr , where d is the product of some short variables. We also will
have some equations among short variables.
The set of all partitions d1 : : : dr corresponding to the same interval Ji can be also
divided into equivalence classes: two such partitions are equivalent if the right (and left)
side variables are equivalent. Every two distinct partitions in the same equivalence class
imply an equation on the short variables.
For every cutting we can consider the following graph ¡: The vertices are equivalence
classes of partitions of intervals with both side variables long. Two vertices are connected
by an edge if they have the equivalent left or right side variables.
We will prove by induction on i , where i < m that variables participating in cuttings
of rank K1 which don't contain long variables in ranks greater than K2 + 2i can be
Á
Á
expressed in terms of short variables and zk K1 = zk K2 +2k where k = 1; : : : ; i: The case
j = 0 is obvious.
Those of the side variables appearing in the partitions of the intervals induced by the
cuttings in ranks K2 +2i+1; : : : ; K1 , that are cutouts of long variables, do not participate
in the cuttings that are not induced by the cuttings in ranks K2 + 2i + 1; : : : ; K1 , hence
don't have to be considered in rank K2 + 2i, but have to be considered in higher ranks.
The intervals that not induced by the cuttings in ranks K2 + 2i + 1 : : : K1 , in this
Á
rank correspond to the cuttings of ¯nite number of words w in zi K2 +2i .
Consider all possible types of subintervals J in rank K2 + 2i together with possible
partitions J = d1 : : : dr , where all variables except possibly d1; dr are short in ranks
K1 ; : : : ; K2 ¡ 2m ¡ 4n + 1: These types can be obtained if in the formulas above we
Á
consider K1 instead of K2 and K1 +2m+4n¡1 instead of K1 . Then zi K1 will correspond
to z¹i .
We have to consider all possible cases when d2 ; : : : ; dr¡1 are contained inside one variable z¹i (because K is not decreasing), and d1 ; dr are obtained from the longer variables
by cutting out maximal integral powers of all bases older or equal than z¹1. The list is
the following.
Á
J1 = zi K1 ;
Á
J2 = zmK1 in case i = m ¡ 1; n = 0;
40
J3 =
ÁK
1
ÁK 1 z i
ci¡1 zi¡1
ci
ÁK
zi¡11
ÁK
zi 1
J4 = ci¡1 ci
J5 =
;
;
ÁK
Á
zi¡11 zi K1
Á
ci ),
zi K1 (ci¡1
z
ÁK
1
J6 = ci i
Á
and d1 entirely includes zi K1 ;
;
ÁK
¡zi 1
J7 = ci
ÁK
zi 1
z
ÁK
1
z
ÁK
1
i¡1
ci i
ci¡1
ÁK
zi¡11
ÁK
1
¡zi
; and d1 entirely contains ci
;
ÁK
zi 1
ÁK
zi 1
and d1 entirely contains ci ;
J8 = ci ci¡1 ci
The last three types of intervals have place if i = m ¡ 1; n = 0:
ÁK
1
zm
J9 = cm
;
ÁK
J10 =
Á
zi¡11 zi K1 zm ÁK1
ci¡1
ci cm
,
Á
ÁK
¡zi 1
zm ÁK1
and dr entirely contains cm
ÁK
¡zi¡11
;
Á
J11 = zmK1 ci
ci¡1 ; and d1 entirely contains zmK1 :
Á
First, notice that none of the long variables (unless it is zi K1 itself can occur on the
left and on the right of some of these intervals.
The intervals J1; : : : ; J9 may have the same long variable on the right, but then
variables on the left will be from di®erent equivalence classes. Intervals J6 ; J8 may have
z
ÁK
1
the same long left variable d1 , but this means that d1 = ci i d; where d is a product of
short variables. Intervals J4 ; J10 and intervals J6; J9 may have the same long variable on
the left, but then variables on the right will be from di®erent equivalence classes.
If i < m ¡ 1, then this immediately implies that the graph ¡ constructed for the
cutting in rank K2 + 2i is a forest. It is not hard to see that the graph ¡ is a forest even
in case i = m ¡ 1; n = 0. Indeed those partitions of J6 which have the left side variable
equivalent to the one in some partition of J8 don't have the right long side variable.
Then none of the partitions of J10 can have the right side variable equivalent to the one
in the partition of some other interval. So if there were a cycle in the graph ¡ in must
contain an edge, connecting two partitions of J6 and J9 , but such a partition of J9 has
the right long variable which does not appear in the partition of the other intervals.
This implies that we can solve a system of equations in short variables and then
express long variables in terms of J1 ; : : : ; J11 and short variables.
We will skip the case i = m; n 6
= 0 because it is similar. Now we will prove by
induction on i that variables participating in cutting of rank K2 + 2m + 4i can be
Á
Á
Á
expressed in terms of short variables and xj K1 ; yj K1 ; j = 1; : : : ; i; zk K1 ; k = 1; : : : ; m.
Á
Á
Suppose we have already expressed in terms of short variables and xj K1 ; yj K1 ; j =
41
Á
1; : : : ; i ¡ 1; zk K1 ; k = 1; : : : ; m variables participating in cuttings of rank K1 which
don't contain long variables in ranks greater than K2 + 2m + 4i ¡ 4. Consider cutting
of rank K2 + 2m + 4i.
Those of the side variables participating in the intervals induced by the partitions in
ranks K2 + 2m + 4i + 1; : : : ; K1, that are cutouts of long variables, do not participate in
the partitions that are not induced by the partitions in ranks K2 + 2m + 4i + 1; : : : ; K1.
The intervals that not induced by the partitions in ranks K2 + 2m + 4i + 1; : : : ; K1,
Á
in this rank correspond to the partitions of ¯nite number of words w in xi K2 +2m+4i ,
Á
Á
Á
yi K2 +2m+4i and xj K2 +2m+4i yj K2 +2m+4i , where j = 1; : : : ; i ¡ 1:
Consider all possible types of these words w together with possible cuttings d1 ; : : : ; dr ,
where d1 is a long variable in rank K2 + 2m + 4i. These types can be obtained if in the
formulas above we consider K1 instead of K2 and K3 = (4m + 2n ¡ 1)(r + 2) instead of
Á
Á
1
K1 . Then xK
= xi K2 +2m+4i will correspond to a¹i and yiK1 = yi K2 +2m+4i will correspond
i
to ¹bi . We have to consider all possible cases when d2 ; : : : ; dr¡1 are contained inside one
variable a¹j or ¹bj (because neither K nor M is decreasing from rank K1 to rank K2 ), and
d1 , dr are obtained from the longer variables by cutting out maximal integral powers of
all bases older or equal than ¹a1 . The list is the following:
Á
1. w = yi K2 +2m+4i ,
Á
2. w = xi K2 +2m+4i ,
Á
¡Á
Á
3.w = xi K2 +2m+4i yi¡1K2 +2m+4i ; and d1 entirely includes xi K2 +2m+4i ,
Á
¡Á
¡Á
Á
¡Á
4. w = xi K2 +2m+4i yi¡1K2 +2m+4i xi¡1K2 +2m+4i ; and d1 entirely includes xi K2 +2m+4i yi¡1K2 +2m+4i .
¡Á
ÁK2 +2m+4i
¡Á
5. w = xi K2 +2m+4i xi¡1
; and d1 entirely contains xi K2 +2m+4i :
¡Á
ÁK2 +2m+4i ¡ÁK2 +2m+4i
¡Á
ÁK2 +2m+4i
6. w = xi K2 +2m+4i xi¡1
yi¡2
; and d1 entirely contains xi K2 +2m+4i xi¡1
:
¡ÁK2 +2m+4i ÁK2 +2m+4i ¡ÁK2 +2m+4i ¡ÁK2 +2m+4i
7. w = xi
xi¡1
yi¡2
xi¡2
;
¡ÁK2 +2m+4i ÁK2 +2m+4i ¡ÁK2 +2m+4i
and d1 entirely contains xi
xi¡1
yi¡2
.
ÁK2 +2m+4i ÁK2 +2m+4i
ÁK2 +2m+4i
8. w = xi
xi¡1
; and d1 entirely contains xi
:
ÁK2 +2m+4i ÁK2 +2m+4i ¡ÁK2 +2m+4i
Á
ÁK2 +2m+4i
9. w = xi
xi¡1
yi¡2
and d1 entirely contains xi K2 +2m+4i xi¡1
:
ÁK2 +2m+4i ÁK2 +2m+4i ¡ÁK2 +2m+4i ¡ÁK2 +2m+4i
10. w = xi
xi¡1
yi¡2
xi¡2
;
ÁK2 +2m+4i ÁK2 +2m+4i ¡ÁK2 +2m+4i
and d1 entirely contains xi
xi¡1
yi¡2
:
ÁK2 +2m+4i
We can cut d1 , express by induction variables obtained in cuttings of xi¡1
and
ÁK2 +2m+4i
ÁK2 +2m+4i
ÁK2 +2m+4i
xi¡1
in terms of xi¡1
and yi¡1
and constants, and then ¯nd d1 for each
of our cases.
In the cases k and k + 1, where k = 2; 3; 5; 6; 8; 9 d1 can be the same, but this
42
means that d1 is just the product of variables it contains entirely, variables obtained
ÁK2 +2m+4i
ÁK2 +2m+4i
in partitions of xi¡1
and yi¡1
and short variables. So in this case we can
disregard this partition and instead we have a partition of xi¡1; yi¡1 with one short side
variable.
Á
Á
Á
If xi K2 +2m+4i and yi K2 +2m+4i both begin with the same d1 then yi K2 +2m+4i ends on a
Á
long dr which does not appear in xi K2 +2m+4i . Hence ¡ is a forest again.
After we solved equations in short variables, the equations in rank K2 + 2i + 2m have
the form yÁK1 = d1d01d2 : : : dr¡1 d0r dr = d1 d~01 d~2 : : : d~r¡1 d~0r dr ; and we either already have
equalities
d01 d2 : : : dr¡1 d0r = d~01 d~2 : : : d~r¡1 d~0r ; where d~01 ; d~2 ; : : : ; d~r¡1 ; d~0r are constants, or
d01 d2 : : : dr¡1 d0r dr = d~01 d~2 : : : d~r¡1 d~0r dr and dr is expressed in terms of the constants and
xÁK2 +2m+4j , yÁK2 +2m+4j , where j < i, and z ÁK2 +k or
d1 d01d2 : : : dr¡1 d0r = d1 d~01 d~2 : : : d~r¡1 d~0r and d1 is expressed in terms of the constants and
ÁK2 +2m+4j
x
, yÁK2 +2m+4j , where j < i, and z ÁK2 +k .
In the ¯rst case we can take an arbitrary dr and put d1 = yÁK1 (d01d2 : : : dr¡1 d0r dr )¡1,
in the second case we put d1 = yÁK1 (d01 d2 : : : dr¡1d0r dr )¡1 , in the third case we put
dr = (d1 d01 d2 : : : dr¡1d0r )¡1 yÁK1 :
2
If we bound all the solutions of the linear systems §M ; : : : ; §K1 +1, by number P ,
then all the elements of the core part of X M can be expressed in terms of the original
variables xÁi M ; yiÁM ; ziÁM . M should be taken large enough, to have K1 > 2m + 4n.
Let °i be an endomorphism of GS de¯ned for GS and X the same way as Ái is
¹ then we write X
¹ ±T = X: We have
de¯ned for G and solution ¯0 of S = 1. If X °T = X,
±T
K1
X = XM for the exponents in ±T bounded by P .
Then all the bases of the rank M cutting can be also expressed in terms of the
original variables xÁi M ; yiÁM ; ziÁM . Taking the same formulas for any solution x~i ; y~i ; z~i we
have a formula solution, because xÁi M ; yiÁM ; ziÁM gives a discriminating solution set.
Notice that we only changed exponential parts of the solution.
If we begin the described process from the solution of the equation vj (X; Y; A) = 1
which satisfy the inequality wj (X; Y; A) 6
= 1, then the constructed formula solution will
also satisfy the inequality wj (X; Y; A) 6
= 1.
Consider now the equation
S=
m
Y
i=1
zi¡1 ci zi
n
Y
x2i = c1 : : : cm
i=1
n
Y
a2i ;
(26)
i=1
where ai ; cj give a solution in a general position (in all the cases when it exists). Let us
43
construct a group G(U; T ) which is obtained from G = F2 by extensions of centralizers
and de¯ne a G-homomorphism à : GS ! G(U; T ) the following way.
In the case m 6
= 0 de¯ne xi µ0 = xi , yi µ0 = yi , zi µ0 = zi and H0 = G: De¯ne H2m¡1
and µ2m¡1 as for equation (14).
µ2m¡1
2m¡1
x2µ
De¯ne G0 = H2m =< H2m¡1 ; r2m j[CH2m¡1 (czmm
); r2m ] = 1 >, and de¯ne
1
¯0
µ2m
r2m
¯0
µ2m
¯0
µ2m¡1
zm = zm = zm r2m ; x1 = x1 = x1 ; and s = s
for all the other variables.
Suppose G2i and ¯2i have been already de¯ned (n ¡ 1 · i · 0). If i < n ¡ 1
¯2i+1
2i+1
2i
de¯ne G2i+1 =< G2i ; t2i+1 j[CG2i ((xi+1xi+2 )¯2i ; t2i+1 ] = 1 >; xi+1
= x¯i+1
t2i+1 ; x¯i+2
=
¯2i
¡1
t2i+1xi+2; and ¯2i+1 acts identically on all the other variables.
2¯
G2i+2 =< G2i+1; t2i+2 j[CG2i+1 (x2¯2i+1
xi+22i+1 ; t2i+2] = 1 >;
i+1
¯2i+2
¯2i+1 t2i+2
¯2i+2
¯2i+1 t2i+2
xi+1
= xi+1
; xi+2
= xi+2
; and ¯2i+2 acts identically on all the other
variables.
In the case i = n ¡ 1 de¯ne ° = ¯2n¡1:
The following result follows from [15].
Proposition 7 The homomorphism ° : GS ! G(U; T ) is an embedding.
We will call, as in the case of equation (14), the corresponding homomorphisms
from G(U; T ) to G standard. A restriction on GS of any positive unbounded family of
standard G-homomorphisms from G(U; T ) onto G, is a discriminating family.
Let K1 = (2n + 2m ¡ 1)(r + 1) and K2 = (2n + 2m ¡ 1)r.
Lemma 12 Suppose that after 2m + 2n steps, when we go from rank
Á
Á
K1 to K2 neither N nor K is decreasing. Denote ¹ai = xi K2 ; c¹i = zi K2 and consider ¹ai ; c¹i as constants. Then the variables corresponding to cutting in rank K1 can be
Á
Á
expressed in terms of xi K1 ; ci K1 and a¹i ; c¹i .
Proof.
Á
Denote a~i = xi K2 +2m+2i¡2 , then
Á
xi K2 +2m+2i¡1 = a~i (~ai a¹i+1 )n1 ;
ÁK2 +2m+2i¡1
xi+1
= (~
ai ¹ai+1)¡n1 ¹ai+1;
2 2
n
Á
xi K2 +2m+2i = (~ai (~
ai a¹i+1 )n1 )(~ai a¹i+1 ) 2 ;
2 2
n
ÁK2 +2m+2i
xi+1
= ((~ai a¹i+1 )¡n1 a¹i+1 )(~ai ¹ai+1 ) 2 ; ³
´n3
2 2
n
2 2
n
ÁK2 +2m+2i+1
xi+1
= ((~
ai a¹i+1 )¡n1 a¹i+1 )(~ai a¹i+1 ) 2 ((~
ai a¹i+1 )¡n1 ¹ai+1 )(~ai a¹i+1 ) 2 a¹i+2
;
Á
³
2 2
n
K2 +2m+2i+1
xi+2
= ((~
ai ¹ai+1 )¡n1 ¹ai+1)(~ai a¹i+1 ) 2 a¹i+2
ÁK2 +2m+2i+2
xi+1
=
44
´¡n3
a¹i+2 ;
³
¡n1
((~ai ¹ai+1)
Á
(~
a2i a
¹2i+1 )n2
¹ai+1)
K2 +2m+2i+2
xi+2
=
µ³
³
¡n1
((~ai ¹ai+1)
2 2
(~
a2i a
¹2i+1 )n2
a¹i+1)
n
((~ai a¹i+1 )¡n1 a¹i+1 )(~ai ¹ai+1 ) 2 a
¹i+2
´¡n3
¹ai+2
¹ai+2
´n4
³
´n3 ´ ((~ai a¹i+1 )¡n1 ¹ai+1 )2(~a2i a¹2i+1 )n2 a¹2
¶
i+2
³
2(~
a2 a
¹2 )n2 2
((~
ai a
¹i+1 )¡n1 ¹
ai+1 ) i i+1
a
¹i+2
´n4
:
Facts 1{4 that have been enumerated in the Proof of Lemma 11 all have place.
The proof is very similar now to the proof of Lemma 11. The possibilities corresponding to the possibilities 1-10 in the proof of Lemma 11 are generated by words
ÁK2 +2m+2i+2 ¡2ÁK2 +2m+2i ¡2ÁK2 +2m+2i¡2
2 +2m+2i+2 ¡2ÁK2 +2m+2i ¡2ÁK2 +2m+2i¡2
xi+1
xi
xi¡1
; and x¡1ÁK
xi
xi¡1
;;
i+1
2ÁK2 +2m+2i 2ÁK2 +2m+2i+2
xi
xi+1
. 2
Theorem 2 has been now proved.
Theorem 3 Suppose we are given a non-exceptional quadratic equation S(x1 ; : : : ; xn ) =
1 and an equation
T (x1; : : : ; xn ; y1 ; : : : ym ; c1; : : : ; ck ) = 1:
Then we can e®ectively ¯nd numbers N and k depending on T = 1 and ¯nite family of
standard N-large homomorphisms fÃ(M1 ;:::;Mk ) jM1; : : : ; Mk ¸ N g from F (U; T ) into F
such that if T = 1 has a solution y1 ; : : : ; ym for all x1 ; : : : ; xn obtained as a restriction
on FS of this family , then T = 1 admits an S-lift.
The proof is basically given in the proof Theorem 2.
Lemma 13 All stabilizing automorphisms (see [11]))of the left side of the equation
cz11 cz22 = c1 c2
(27)
have form
n
z1Á = ck1 z1(c1 c2 )n ; z2Á = cm
2 z2 (c1 c2 ) :
All stabilizing automorphisms of the left side of the equation
x2cz = a2 c
(28)
have form
2 n
xÁ = x(a c) ; z Á = ck z(a2 c)n .
All stabilizing automorphisms of the left side of the equation
x21 x22 = a21 a22
have form
2 2 n
2 2 n
xÁ1 = (x1 (x1x2 )m )(a1 a2 ) ; xÁ2 = ((x1x2 )¡m x2 )(a1 a2 ) :
45
(29)
;
Proof can be done by software \Magnus".
By the result of Grigorchuk, Kurchanov and Lysionok (Theorem 5 in [11]) every
solution of such equation can be obtained from a ¯nite number of basic solutions by
application of stabilizing automorphisms.
Using the same technique one can prove the similar result for equations over ¯nitely
generated subgroup of Lyndon's group.
In the cases when the quadratic equation has only commutative solutions, Rad(S) =
nclf[xi ; bi ] = 1; i = 1; : : : ; kg, where bi 's are constants.
Lemma 14 Suppose the equation u = 1 has the form [x1 ; x2] = 1 Consider a subgroup of FkZ[x] given by the presentation H =< Fk ; t1 ; t2j[t1 ; t2] = 1 >.
Given an equation v = 1 and an inequality w 6
= 1 there exists a ¯nite number of pairs
(k1 ; m1); (k2 ; m2) and embeddings
k2 m2
1
Ã(ki ;mi ) : x1 ! tk11 tm
2 ; x2 ! t1 t2 ;
such that if the formula
© = 8x1 ; x2 (u = 1 ! 9Y (v(x1 ; x2 ; Y; A) = 1
^
w(x1 ; x2 Y; A) 6
= 1)
is true in Fk (a1; : : : am are the ¯rst m generators of Fk ) then one of the formulas
©i = 9Y (v(Á(ki ;mi ) (x1 ); Á(ki ;mi )(x2 ); Y; a; b) = 1
^
w1 (Á(ki ;mi ) (x1 ); Á(ki ;mi )(x2 ); Y; a; b) 6
= 1)
is true in H, and any x1 ; x2 satisfying u = 1 can be obtained from this ¯nite number of
pairs by a specialization of t1; t2.
This lemma follows from Razborov's process applied to v = 1 considering Y as
variables and x1; x2 (or x) as parameters.
6.2
Generalization of Theorem 2 for triangular quasi-quadratic
systems
Theorem 4 Let u[Xn ; : : : ; X1; A] = 1 be a non-degenerate triangular quasi-quadratic
system, such that all the quadratic equations are in the standard form.
Replace each quadratic equation of the form cx1 i1 cx2 i2 = c1c2 ; where [c1; c2 ] 6
= 1, in the
system u by the systems xi1 = z1c¹1 z3; xi2 = z2 c¹2 z3; [z1 ; c1 ] = 1; [z2 ; c2 ] = 1; [z3 ; c1 c2] = 1;
each equation of the form x2i1 cxi2 = a2 c, where [a; c] 6
= 1, by the system xi1 = ¹az1 ; xi2 =
z2c¹z1 ; [z2; c] = 1; [z1 ; a2 c] = 1, each equation of the form x2i1 x2i2 = a21 a22 by the system
xi1 = (¹
a1z1 )z2 ; xi2 = (z1¡1a¹2 )z2 ; [z1 ; a¹1a¹2 ] = 1; [z2 ; a21 a22 ] = 1.
46
Here ¹a; a¹1; a¹2 ; c¹; c¹1 ; c¹2 run through the ¯nite number of solutions of the corresponding
equation over the coordinate group of the system below this equation.
Denote by u1 = 1; : : : ; uk = 1 the new systems.
If the formula
© = 8X1; : : : ; Xn (u = 1 ! 9Y (v(X1 ; : : : ; Xn ; Y; A) = 1
is true in G (we suppose that
^
©1 = 8X1; : : : ; Xn (u1 = 1 ! 9Y (v(X1 ; : : : ; Xn ; Y; A) = 1
w(X1 ; : : : ; Xn ; Y; A) 6
= 1)
^
w(X1; : : : ; Xn ; Y; A) 6
= 1)
is true ) then there are embeddings µi : GR(u1 ) = Gi ! GR(u1 ) , i = 1; : : : t (t depends on
u; v; w), of copies of GR(u1 ) , such that one of the formulas
©i = 9Y (v(µi (X1 ); : : : ; µi (Xn ); Y; A) = 1
is true in the group GR(u) .
^
w(µi (X1 ); : : : ; µi (Xn ); Y; A) 6
= 1)
Suppose the system u = 1 is a union of equations Si (Xi ; : : : ; X1 ; A) = 1, each of
them quadratic in the variables from Xi . Every equation Si = 1 has a solution in
FR(Si¡1 [:::[S1 ) , because the system is non-degenerate.
Suppose the formula © is true in F (a free group of rank m + p with the ¯rst m
generators A).
By theorem 1 one can ¯nd a ¯nite number of schemes (u&v)[Y; Xn ; : : : ; X1 jY1 ; Hn ; : : : ; H1 ] =
1, such that for any solution X of u(X; A) = 1 one of these schemes takes place. Take
some scheme which works for a positive unbounded family of solutions of u = 1. Consider the parts of this scheme corresponding to Xn . Each of them can be represented as
a generalized equation (see [16]) with unknowns (or items) Hn , such that the interval I
consists of the union of the intervals corresponding to Xn = fxn1 ; : : : ; xnl g: Each xij is
cut in several ways into bases ¸ij .
For each Xi we can consider variables from X1 ; : : : ; Xi¡1 as parameters Our goal is
to prove that these ¸ij can be expressed as functions of x 2 X; A:
We proceed by induction.
First consider the parts of the schemes corresponding to X1. We are in the situation
of Theorem 2. Namely, we have some cutting of variables from X1 , which are arbitrary
solutions of a quadratic equation S1 (X1 ) = 1. If the equation has form as in Theorem 2,
then the bases ¸1j by Theorem 2 can be expressed as functions of x 2 X1 ; A. If the equation S1 = 1 has only commutative solutions, then we take as ¸1j their images in FR(S1 )
described in Lemma 14. We know from [15], that FR(Si [:::[S1 ) = (FR(Si¡1 [:::[S1 ) )R(Si ) .
47
Suppose by induction that we have already proved that there is a ¯nite number of
ways to represent the bases ¸1j ; : : : ; ¸i¡1;j as elements of FR(Si¡1 [:::[S1 ) , such that the
generalized equation corresponding the part of the scheme involving only Xi¡1 [ : : : [ X1
is satis¯ed (and inequality w 6
= 1 is satis¯ed) if and only if for images of the bases
Á(¸1j ); : : : ; Á(¸i¡1;j ) in FR(Si¡1 [:::[S1 ) that are given by one of these formulas the equation
v = 1 and some inequality w1;i¡1 6
= 1 is satis¯ed in FR(Si¡1 [:::[S1 ). Take a positive
unbounded family of solutions for Si = 1. By Theorem 2 one can express ¸i;j 's in terms
of x 2 Xi and the bases ¸i¡1;j ; : : : ¸1;j , which play the role of constants for Si (when
in the proof of Theorem 2 we solve equations in constants, these constants are not the
coe±cients of the original quadratic equation, but are possibly obtained by cutting of
these coe±cients, this is why the role of constants is played by ¸i¡1;j ; : : : ¸1;j , and not
by xi¡1;j ; : : : x1;j ).
By induction ¸i¡1;j ; : : : ¸1;j are represented as elements of FR(Si¡1 [:::[S1 ) , hence ¸ij 's
are represented as elements of FR(Si [:::[S1 ) .
We can see that the generalized equation corresponding to the part involving only
Xi [ : : : [ X1 of the scheme is satis¯ed (and the inequality w 6
= 1 is satis¯ed) if and
only if for the images of the bases Á(¸1j ); : : : ; Á(¸i;j ) in FR(Si )[:::[S1 ) that are given by
one of these formulas the equation v = 1 and the new inequality w1;i 6
= 1 is satis¯ed in
FR(Si [:::[S1 ) .
Theorem 4 is proved.
6.3
The Implicit function theorem in general formulation
De¯nition 12 Suppose S = 1 is an NTQ system
S1(X1; X2; : : : ; Xm )
S2(X2; : : : ; Xm )
=1
=1
...
Sm (Xm ) = 1;
where the system Sj = 1 is quadratic in variables from Xj . If Xj = fxj1 ; : : : ; xjk g and all
the equations of the system Sj = 1 have form [xjs ; wjs (Xj+1 ; : : : ; Xm )] = 1; s = 1; : : : k;
and wjs (Xj+1 ; : : : ; Xm ) some words, then the variables from Xj are called exceptional
variables.
48
An embedding µ of FR(S) into itself ¯xing all non-exceptional variables and sending
each exceptional variable x into some element from the centralizer CFR(S) (x), is called a
corrective embedding.
Theorem 5 [Implicit function theorem] Suppose that S[Xk ; Yk¡1; : : : ; X1 ] = 1 is a
nondegenerate triangular quasi-quadratic system of the following form:
Sk [Xk ; Yk¡1; : : : ; X1 ] = 1;
Rk¡1[Yk¡1 ; : : : ; X1 ] = 1;
Sk¡1[Xk¡1 ; Yk¡2; : : : ; X1 ] = 1;
:::::
R1 [Y1 ; X1 ] = 1;
S1[X1 ] = 1;
where Sk = 1; : : : ; S1 = 1 are triangular quasi-quadratic systems (generalized systems),
in variables Xk ; : : : ; X1 respectively, where exceptional equations from Theorem 4 are
replaced by corresponding systems from Theorem 4, and Rk¡1 = 1; : : : ; R1 = 1 are
triangular quasiquadratic systems in variables Yk¡1 ; : : : ; Y1 respectively.
Suppose
® = 8X19Y1 : : : 8Xk 9Yk (S = 1 ! v(X1 ; Y1; : : : ; Xk ; Yk ) = 1):
is true in F . Then there exist a scheme (a generalized scheme) T [Yk ; Xk ; : : : ; Y1 ; X1 jYk(1) ; Xk ; : : : ;
1 for a system S(µ(X1 ); Y1 ; : : : ; µ(Xk ); Yk ) = 1&v(µ(X1 ); Y1 ; : : : ; µ(Xk ); Yk ) = 1, where µ
is some corrective embedding of FR(S) into itself, with triangular quasi-quadratic system
in the following form
(1)
(1)
Pk [Yk ; Xk ; Yk¡1 : : : ; X1 ] = 1;
Sk [Xk ; Yk¡1; : : : ; X1 ] = 1;
:::::
P2 [Y2(1) ; X2 ; Y1(1) ; X1 ]
= 1;
S2[X2 ; Y1 ; X1 ] = 1;
(1)
P1 [Y1 ; X1 ] = 1;
S1[X1 ] = 1;
and Yi = p~i (Yi(1)).
49
Proof Consider some complete family of schemes for S = 1& _rj=1 vj = 1, say T1 =
1; : : : ; Tt = 1: Then by Lemma 8 in [15] and by Theorem 4 there is a ¯nite number of
i
substitutions Xi sm
! Xi in T1 ; : : : ; Tt ; such that for some of these substitutions and
i
for some positive unbounded family of solutions of equations S1 = 1; : : : ; Sk = 1 there
exists one of the schemes, say
T1 [Yk ; Xk ; : : : ; Y1 X1 jYk(1); Xk(1) ; : : : ; Y1(1) X1(1) ] = 1
such that for X's in this family Y 's can be determined by T1 = 1. Then by Theorem 2 each element of Yk(1) can be expressed in terms of Xk and some pieces of
Yk¡1 ; Xk¡1; : : : ; X1 (which corresponds to short Yk 's). We will make the following trans(1)
formation with the generalized equation corresponding to v1 = 1: suppose xi =
(1)
(1)
xi1 : : : xit ; if ¹ = xi is a base in the family Xk and the double base ¢(¹) belongs
(1)
to Yk . Then we cut ¢(¹) into t parts: ¢(¹1 ); : : : ; ¢(¹t ), delete ¹, and for each
(1)
j = 1; : : : ; t draw the base ¹j on xij : The Yk -part will still be quadratic, and the
(1)
elements of Xk will not be cut anymore. If some base ¸ was on Xk and ¢(¸) was on
(1)
Yk¡1 , then it corresponds to a short variable from Yk and is moved to some base on
Yk by the procedure described above. Then repeat this procedure for Yk¡1 and so on.
We obtain the scheme T = 1, which has Yj parts similar to the scheme T1 = 1, only
(1)
constant bases on Yj(1) -parts are changed and represented as words in Xj ; Yj¡1
: : : ; X1 :
Changing parameters in these words corresponds to taking di®erent elements in positive
unbounded family for X's.
There is a ¯nite number of such schemes corresponding to the in¯nite family of values
for integer parameters in the positive unbounded family. In these schemes there are new
bases corresponding to the analogs of letters ts;i ; rs;i for s > K1 on each step. 2
Theorem 6 Suppose
® = 8X1 9Y1 : : : 8Xk 9Yk u = 1 ! _rj=1 (vj (X1 ; Y1 ; : : : ; Xk ; Yk ) = 1&wj (X1 ; Y1; : : : ; Xk ; Yk ) 6
= 1)
is true in F , and u1 = 1; : : : ; uq = 1 are schemes for u = 1. Then for some i 2 f1; : : : ; qg
there are functions f1 ; : : : ; fn , and a corrective substitution µ, such that
FR(ui (X1 ;f1 (X1 );X2 ;f2 (X1 ;X2 );:::)) ` _rj=1(vj (µ(X1); f1 (X1 ); : : : ; µ(Xk ); fk (X1; : : : Xk )) = 1
&wj (µ(X1 ); f1(X1); : : : ; µ(Xk ); fk (X1 ; : : : Xk )) 6
= 1):
50
References
[1] K.I.Appel. One-variable equations in free groups. Proc. Amer. Math. Soc., 19:912{
918, 1968.
[2] G. Baumslag, A. Myasnikov, and V. Remeslennikov. Algebraic geometry over
groups I: Algebraic sets and ideal theory. 1999, to appear.
[3] Baumslag G., Myasnikov A., Remeslennikov V., Residually hyperbolic groups. Proc.
Inst. Appl. Math. Russian Acad. Sci., 1995, 24, p.3-37.
[4] Baumslag G. On generalized free products. Math. Zeitschr., 7, 8: 423{438, 1962.
[5] Bryant R. The verbal topology of a group. Journal of Algebra, 48:340{346, 1977.
[6] C.C. Chang and H.J. Keisler. Model Theory. North-Holland, London, N.Y, 1973.
[7] L.P. Comerford and C.C. Edmunds. Quadratic equations over free groups and free
products. Journal of Algebra, 68:276{297, 1981.
[8] L. P. Comerford Jr. and C. C. Edmunds. Solutions of equations in free groups.
Walter de Gruyter, Berlin, New York, 1989.
[9] Yu. L. Ershov and E. A. Palutin. Mathematical Logic. Walter de Gruyter, Berlin,
New York, 1989.
[10] R. I. Grigorchuk and P. F. Kurchanov. Some questions of group theory connected
with geometry. In Encyclopedia of mathematics, 1989.
[11] R.I. Grigorchuk and P.F. Kurchanov. On quadratic equations in free groups. Contemp. Math., 131(1):159{171, 1992.
[12] Guba V. Equivalence of in¯nite systems of equations in free groups and semigroups
to ¯nite subsystems. Mat. Zametki, 40:321{324, 1986.
[13] A. Hoare, A. Karrass, D.Solitar. Subgroups of ¯nite index of Fuchsian groups.
Math. Z., 120:289{298, 1971.
[14] A. Hoare, A. Karrass, D.Solitar. Subgroups of ¯nite index of Fuchsian groups.
Math. Z., 125:59{69, 1972.
51
[15] O. Kharlampovich and A. Myasnikov. Irreducible a±ne varieties over a free group.
1: irreducibility of quadratic equations and Nullstellensatz. J. of Algebra, 200:472{
516, 1998.
[16] O. Kharlampovich and A. Myasnikov. Irreducible a±ne varieties over a free group.
2: systems in triangular quasi-quadratic form and description of residually free
groups. 1998.
[17] A.A. Lorenc. The solution of systems of equations in one unknown in free groups.
Dokl. Akad. Nauk SSSR, 4:262{266, 1963.
[18] R.C. Lyndon. The equation a2b2 = c2 in free groups. Michigan Math. J.., 6:155{164,
1959.
[19] R.C. Lyndon. Equations in free groups. Trans. Amer. Math. Soc., 96:445{457, 1960.
[20] R.C. Lyndon. Groups with parametric exponents. Trans. Amer. Math. Soc., 96:518{
533, 1960.
[21] R.C. Lyndon. Length functions in groups. Math. Scand., 12:209{234, 1963.
[22] R.C. Lyndon. Equations in groups. Bol. Soc. Bras. Mat., 11:79{102, 1980.
[23] Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory. Springer, 1977.
[24] A.I.Malcev Decidability of the universal and positive theories of a free group (Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 48(1):735{749, 1985. transl. in Math.
USSR Izv., V. 25, 1985; MR 86c:03009.
[25] G.S. Makanin. Decidability of the universal and positive theories of a free group
(Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 48(1):735{749, 1985. transl. in Math.
USSR Izv., V. 25, 1985; MR 86c:03009.
[26] G.S. Makanin. Equations in a free group (Russian). Izv. Akad. Nauk SSSR, Ser.
Mat., 46:1199{1273, 1982. transl. in Math. USSR Izv., V. 21, 1983; MR 84m:20040.
[27] A. I. Mal'cev. On some correspondence between rings and groups. Math. Sbornik,
50):257{266, 1960.
[28] Ju. I. Merzljakov. Positive formulae on free groups. Algebra i Logika, 5(4):25{42,
1966.
52
[29] A. G. Myasnikov and V. N. Remeslennikov. Exponential groups 2: extension of
centralizers and tensor completion of csa-groups. International Journal of Algebra
and Computation, 6(6):687{711, 1996.
[30] B. Plotkin Varieties of algebras and algebraic varieties. Categories of algebraic
varieties. Preprint, Hebrew University, Jerusalem, 1996.
[31] W. Quine. Concatenation as a basis for arithmetic.. J. of Symb. Logic, 11:105{114,
1946.
[32] A. Razborov. On systems of equations in a free group. Math. USSR, Izvestiya,
25(1):115{162, 1985.
[33] A. Razborov. On systems of equations in a free group. PhD thesis, Steklov Math.
Institute, Moscow, 1987.
[34] A. Razborov. On systems of equations in free groups. In Combinatorial and geometric group theory. Edinburgh 1993, pages 269{283, Cambridge University Press,
1995.
[35] Remeslennikov V.N. E{free groups. Siberian Math. J., v.30, 6:153{157, 1989.
[36] G.S. Sacerdote. Elementary properties of free groups. Trans. of the AMS, 178,
1973.
[37] Szmielew W. Elementary properties of Abelian groups. Fund. Math., 41:203{271,
1955.
[38] J.R. Stallings. Finiteness of matrix representation. Ann. Math., 124:337{346, 1986.
[39] H. Bass. Groups acting on non-archimedian trees. 1991.
[40] D. Gildenhuys, O. Kharlampovich and A. Myasnikov. CSA groups and separated
free constructions. Bull. Austral. Math. Soc.,, 52: 63{84, 1995.
53