A weak Zassenhaus lemma for discrete
subgroups of Diff(I)
Azer Akhmedov
ABSTRACT: We prove a weaker version of the Zassenhaus Lemma for subgroups of Diff(I). We also show that a group with commutator subgroup containing
a non-abelian free subsemigroup does not admit a C0 -discrete faithful representation in Diff(I).
In this paper, we continue our study of discrete subgroups of Diff + (I)
- the group of orientation preserving diffeomorphisms of the closed interval I = [0, 1]. Following recent trends, we try to view the group
Diff + (I) as an analog of a Lie group, and we study still basic questions about discrete subgroups of it. This paper can be viewed as a
continuation of [A] although the proofs of the results of this paper are
independent of [A].
Throughout the paper, the letter G will denote the group Diff + (I).
On G, we assume the metric induced by the standard norm of the
Banach space C 1 [0, 1]. We will denote this metric by d1 . Sometimes,
we also will consider the metric on G that comes from the standard
sup norm ||f ||0 = sup x∈[0,1] |f (x)| of C[0, 1] which we will denote by
d0 . However, unless specified, the metric in all the groups Diff r+ (I), r ∈
R, r ≥ 1 will be assumed to be d1 .
The central theme of the paper is the Zassenhaus Lemma. This
lemma states that in a connected Lie group H there exists an open nonempty neighborhood U of the identity such that any discrete subgroup
generated by elements from U is nilpotent (see [R]). For example, if H
is a simple Lie group (such as SL2 (R)), and Γ ≤ H is a lattice, then Γ
cannot be generated by elements too close to the identity.
In this paper we prove weak versions of the Zassenhaus Lemma for
the group G = Diff + (I). Our study leads us to showing that finitely
generated groups with exponential growth which satisfy a very mild
condition do not admit faithful C0 -discrete representation in G:
Theorem A. Let Γ be a subgroup of G, and f, g ∈ [Γ, Γ] such
that f and g generate a non-abelian free subsemigroup. Then Γ is not
C0 -discrete.
1
2
We also study the Zassenhaus Lemma for the relatives of G such as
Diff 1+c
+ (I), c ∈ R, c > 0 - the group of orientation preserving diffeomorphisms of regularity 1 + c. In the case of Diff 1+c
+ [0, 1], combining
Theorem A with the results of —bf [N2], we show that C0 -discrete
subgroups are more rare.
Theorem B. Let Γ be a C0 -discrete subgroup of Diff 1+c
+ [0, 1]. Then
Γ is solvable with solvability degree at most k(c).
Theorem B can be strengthened if the regularity is increased further;
combining Theorem A with the results of Navas [N2], Plante-Thurston
[PT], and Szekeres [S] we obtain the following
Theorem C. If Γ is C0 -discrete subgroup of Diff 2+ [0, 1] then Γ is
metaabelian.
It follows from the results of [A], as remarked there, that the Zassenhaus Lemma does not hold either for Diff + (I) or for Homeo+ (I), in
metrics d1 and d0 respectively.
In the increased regularity the lemma still fails: given an arbitrary
open neighborhood U of the identity diffeomorphism in G, it is easy to
find two C ∞ “bump functions” in U which generate a discrete group
isomorphic to Z o Z; thus the lemma fails for Diff ∞
+ (I).
Because of the failure of the lemma, it is natural to consider strongly
discrete subgroups which we have defined in [A]. Indeed, for strongly
discrete subgroups, we are able to obtain positive results which are
natural substitutes for the Zassenhaus Lemma.
Let us recall the definition of strongly discrete subgroup from [A]:
Definition 1. Let Γ be a subgroup of Diff + (I). Γ is called strongly
discrete if there exists C > 0 and x0 ∈ (0, 1) such that |g 0 (x0 ) − 1| >
C for all g ∈ Γ\{1}. Similarly, we say Γ is C0 -strongly discrete if
|g(x0 ) − x0 | > C for all g ∈ Γ\{1}.
Let us note that a strongly discrete subgroup of G is discrete; and
C0 -strongly discrete subgroup of G is C0 -discrete.
For convenience of the reader, let us recall several basic notions on
the growth of groups: if Γ is a finitely generated p
group, and S a finite
generating set, we will define ω(Γ, S) = limn→∞ n |Bn (1; S, Γ)| where
Bn (1; S, Γ) denotes the ball of radius n around the identity element.
(Often we will denote this ball simply by Bn (1)). We will also write
ω(Γ) = inf |S|<∞,hSi=Γ ω(Γ, S) where the infimum is taken over all finite
3
generating sets S of Γ. If ω(Γ) > 1 then one says that Γ has uniform
exponential growth.
Now we are ready to state weak versions of the Zassenhaus Lemma
for the group G. First, we state a theorem about C0 -strongly discrete
subgroups.
Theorem 1. Let ω > 1. Then there exists an open non-empty
neighborhood U of the identity 1 ∈ Diff 1+ [0, 1] such that if Γ is a finitely
generated C0 -strongly discrete subgroup of Diff 1+ [0, 1] with ω(Γ) ≥ ω,
then Γ cannot be generated by elements from U .
By increasing the regularity, we can prove a similar version for strongly
discrete subgroups
Theorem 2. Let ω > 1. Then there exists an open non-empty
neighborhood U of the identity 1 ∈ Diff 1+ [0, 1] such that if Γ is a finitely
generated strongly discrete subgroup of Diff 2+ [0, 1] with ω(Γ) ≥ ω, then
Γ cannot be generated by elements from U .
Remark 1. In regard to the Zassenhaus Lemma, it is interesting
to ask a reverse question, i.e. given an arbitrary open neighborhood U
of the identity in G, is it true that any finitely generated torsion free
nilpotent group Γ admits a faithful discrete representation in G generated by elements from U ? In [FF], it is proved that any such Γ does
admit a faithful representation into G generated by diffeomorphisms
from U . Also, it is proved in [N1] that any finitely generated nilpotent
subgroup of G indeed can be conjugated to a subgroup generated by
elements from U .
Remark 2. Because of the assumptions about uniform exponential
growth in Theorem 1 and Theorem 2, it is natural to ask whether or
not every finitely generated subgroup of G of exponential growth has
uniformly exponential growth. This question has already been raised
in [N2].
Acknowledgement. We are thankful to Andrés Navas for his remarks,
and for bringing [N1] to our attention. We also would like to thank to
two anonymous referees for many helpful remarks and suggestions.
Proofs of Theorem 1 and Theorem 2
Proof of Theorem 1. We can choose λ > 1 such that λ < ω(Γ).
Then the cardinality of the sphere of radius n of Γ with respect to any
4
fixed finite generating set is bigger than the exponential function λn ,
for infinitely many n.
Then let > 0 such that (1 − 10)λ > 1. We let U be the neighborhood of the identity in G with respect to d1 metric (we always
assume d1 metric in G unless otherwise stated).
Let Γ be generated by finitely many non-trivial diffeomorphisms
f1 , f2 , . . . , fs ∈ U . We fix this generating set and denote it by S,
i.e. S = {f1 , f1−1 , . . . , fs , fs−1 }.
We want to prove that Γ is not C0 -strongly discrete. Assuming the
opposite, let x0 ∈ (0, 1) such that for some C > 0, |g(x0 ) − x0 | > C
for all g ∈ Γ\{1}.
Let Bn (1) be the ball of radius n around the identity in the Cayley
graph of Γ with respect to S. Then Card(Bn (1)\Bn−1 (1)) > λn for
infinitely many n ∈ N. Let A denotes the set of all such n.
Let ∆ be a closed subinterval of (0, 1) of length less than C such that
x0 is the left end of ∆.
We denote the right-invariant Cayley metric of Γ with respect to S
by |.|. For all g ∈ Γ, let ∆g = g(∆). Thus we have a collection {∆g }g∈G
of closed subintervals of (0, 1).
Notice that if g = sw, s ∈ S then by Mean Value Theorem,
P |∆sw | >
(1−10)|s(∆w )|. Then, necessarily, for all n ∈ A, we have |g|=n |∆g | >
(1 − 10)n λn |∆| → ∞ as n → ∞.
Then ∃g1 , g2 ∈ Γ, g1 6= g2 such that g2 (x0 ) ∈ ∆g1 . Then g1−1 g2 (x0 ) ∈
∆. Since |∆| < C we obtain a contradiction. Now we prove a better result by assuming higher regularity for the
representation
Proof of Theorem 2. Let λ, λ1 , λ2 be constants such that 1 <
λ < λ1 < λ2 < ω(Γ). Then the cardinality of the sphere of radius n
of Γ with respect to any fixed finite generating set is bigger than the
exponential function λn2 , for infinitely many n.
λ
We choose > 0, η > 0 to be such that 1 < η < 1+
and 1+
<
1−
Let U be the ball of radius around the identity diffeomorphism.
λ1
.
λ
We again assume Γ is generated by finitely many non-trivial diffeomorphisms f1 , f2 , . . . , fs ∈ U , and we fix the generating set S =
{f1 , f1−1 , . . . , fs , fs−1 }. Let Bn (1) be the ball of radius n around the
identity in the Cayley graph of Γ with respect to S. Then we have
5
Card(Bn (1)\Bn−1 (1)) > λn2 for infinitely many n ∈ N. Let A denotes
the set of all such n.
We need to show that Γ is not strongly discrete. Assuming the
opposite, let x0 ∈ (0, 1) such that for some C > 0, |g 0 (x0 ) − 1| > C for
all g ∈ Γ\{1}.
Let C1 be a positive number such that
1 − C < (1 − C1 )2 and 1 + C > (1 + C1 )2
Let also N1 ∈ N such that for any n ≥ N1 , we have
1
1
1 − C1 < (1 − n )n < (1 + n )n < 1 + C1
η
η
Notice that for all n ∈ A, g ∈ Bn (1)\Bn−1 (1), and x ∈ [0, 1], we have
1+
< λλ1 , there exists n ∈ A and
(1 − )n < g 0 (x) < (1 + )n . Since 1−
g1 , g2 ∈ Γ such that
n > N1 , g1 6= g2 , |g1 | = |g2 | = n
but
|g1 (x0 ) − g2 (x0 )| ≤
1
g10 (x0 )
(?
),
and
1
−
C
<
< 1 + C1 (?2 )
1
1
λn
g20 (x0 )
Indeed, by the pigeonhole principle, for all n ∈ A, there exists j ∈
{0, 1, . . . , [λn ]} such that
λn
j j+1
Card{g ∈ Bn (1)\Bn−1 (1) | g(x0 ) ∈ [ n , n )} ≥ n 2
λ
λ
λ +1
For all n ∈ A, j ∈ {0, 1, . . . , [λn ]}, let
j j+1
,
)}
λn λn
Then, for sufficiently big n ∈ A, there exists j ∈ {0, 1, . . . , [λn ]} such
λn
that Card(D(n, j)) ≥ λ1n (?3 ). For all n ∈ A, let
D(n, j) = {g ∈ Bn (1)\Bn−1 (1) | g(x0 ) ∈ [
J(n) = {j ∈ {0, 1, . . . , [λn ]} | Card(D(n, j)) ≥
λn1
}
λn
Recall also that for all g ∈ D(n, j), we have
(1 − )n < g 0 (x0 ) < (1 + )n
1+
Then, since 1−
< λλ1 , for sufficiently big n ∈ A and j ∈ J(n), applying
the pigeonhole principle to the set D(n, j), we obtain that (besides
the inequality (?3 )) there exist distinct g1 , g2 ∈ D(n, j) such that the
g 0 (x )
inequality 1 − C1 < g10 (x00 ) < 1 + C1 holds. On the other hand, by
2
6
definition of D(n, j), we have |g1 (x0 )−g2 (x0 )| ≤
the desired inequalities (?1 ) and (?2 ).
1
;
λn
thus we established
Let now y0 = g1 (x0 ), z0 = g2 (x0 ), W = g1−1 , V = g1−1 g2 , and let
W = hn hn−1 . . . h1 where W is a reduced word in the alphabet S of
length n and hi ∈ S, 1 ≤ i ≤ n.
Let also Wk be the suffix of W of length k, yk = Wk (y0 ), zk =
Wk (z0 ), 1 ≤ k ≤ n.
Furthermore, let max1≤i≤s sup0≤y6=z≤1
|fi0 (y)−fi0 (z)|
|y−z|
= M , and L = 1 + .
Then we have
|yk − zk | ≤
Lk
M Lk
0
0
,
|h
(y
)
−
h
(z
)|
≤
,0 ≤ k ≤ n − 1
k+1 k
k+1 k
λn
λn
k+1
h0
(y )
k+1
k
≤ 1 + M Lλn for all 0 ≤ k ≤ n − 1. From
Then 1 − M Lλn ≤ hk+1
0
(z
)
k
k+1
here we obtain that
h0k+1 (yk )
M Lk+1
M Lk+1
n−1
n
Πn−1
(1
−
)
≤
Π
≤
Π
(1
+
)
k=0 0
k=0
k=0
λn
hk+1 (zk )
λn
Then, for sufficiently big n in A
0
1 n
1 n
1
1
n−1
n−1 hk+1 (yk )
≤ Πn−1
(1 − n ) = Πk=0 (1 − n ) ≤ Πk=0 0
k=0 (1 + n ) = (1 + n )
η
η
hk+1 (zk )
η
η
h0
(y )
n−1 k+1 k
Since, by the chain rule, Πk=0
=
h0 (zk )
k+1
(g1−1 )0 (y0 )
,
(g1−1 )0 (z0 )
we obtain that
(g1−1 )0 (y0 )
1 − C1 < −1 0
< 1 + C1
(g1 ) (z0 )
Then
0
V (x0 ) =
(g1−1 )0 (g2 (x0 ))g20 (x0 )
(g1−1 )0 (g2 (x0 ))g20 (x0 ) −1 0
(g1 ) (g1 (x0 ))g10 (x0 ) =
= −1 0
0
(g1 ) (g1 (x0 ))g1 (x0 )
(g1−1 )0 (g2 (x0 ))g20 (x0 )
(g1−1 )0 (g2 (x0 )) g20 (x0 )
∈ ((1−C1 )2 , (1+C1 )2 ) ⊂ (1−C, 1+C)
= −1 0
−1 0
0
0
(g1 ) (g1 (x0 ))g1 (x0 )
(g1 ) (g1 (x0 )) g1 (x0 )
Thus we proved that 1 − C < V 0 (x0 ) < 1 + C which contradicts our
assumption. Remark 3. The same proof, with slight changes, works for representations of C 1+c -regularity for any real c > 0.
7
Proofs of Theorems A, B, C
In the proofs of Theorem 1 and of Theorem 2, we consider the orbit
of the point x0 under the action of Γ. By using exponential growth,
we find two distinct elements g1 , g2 such that g1 (x0 ) and g2 (x0 ) are
very close. Then we “pull back” g2 (x0 ) by g1−1 , i.e. we consider the
point g1−1 g2 (x0 ) and show that this point is sufficiently close to x0 . It
is at this stage that we heavily use the condition that Γ is generated
by elements from the small neighborhood of 1 ∈ G, i.e. derivatives of
the generators are uniformly close to 1. However, if Γ is an arbitrary
subgroup of the commutator group [G, G], not necessarily generated by
elements close to the identity element, then for any x0 ∈ (0, 1), f ∈ Γ
and for any > 0, there exists W ∈ Γ such that |f 0 (W (x0 )) − 1| < ;
we simply need to find W such that W (x0 ) is sufficiently close to 1
(or to 0). This fact provides a new idea of taking x0 close to 1, then
considering the part of the orbit which lies in a small neighborhood of
1, then using exponential growth to find points close to each other in
that neighborhood, and then perform the “pull back”.
The following proposition is a special case of Theorem A, and answers
Question 2 from [A]. For simplicity, we give a separate proof of it.
Proposition 1. F2 does not admit a faithful C0 -discrete representation in G.
Proof. Since the commutator subgroup of F2 contains an isomorphic
copy of F2 , it is sufficient to prove that F2 does not admit a faithful
C0 -discrete representation in G(1) = [G, G].
Let Γ be a subgroup of G(1) isomorphic to F2 generated by diffeomorphisms f and g. Without loss of generality we may assume that Γ has
no fixed point on (0, 1). Let also > 0 and M = max0≤x≤1 (|f 0 (x)| +
|g 0 (x)|).
√
We choose N ∈ N, δ > 0 and θN such that 1/N < , 1 < θN < 2N 2,
and for all x ∈ [1 − δ, 1], the inequality θ1N < φ0 (x) < θN holds where
φ ∈ {f, g, f −1 , g −1 }.
Let W = W (f, g) be an element of Γ such that W (1/N ) ∈ [1 − δ, 1],
m be the length of the reduced word W . Let also xi = i/N, 0 ≤ i ≤ N .
For every n ∈ N, let
Sn = {H ∈ Bn (1) | u(W (x1 )) ≥ W (x1 ) for all suffixes u of H}
(Here we view H as a reduced word in the alphabet {f, g, f −1 , g −1 }).
Then |Sn | ≥ 2n .
8
Then (assuming N ≥ 3) we can choose and fix a sufficiently big n
such that the following two conditions are satisfied:
(i) there exist g1 , g2 ∈ Sn such that g1 6= g2 , and
1
√ n , 1 ≤ i ≤ N − 1.
|g1 W (xi ) − g2 W (xi )| < 2N
2
(ii) M m (θN )n
1
√ n
2N
2
< .
Indeed,
numbers such
√ let (c0 , c1 , . . . , cN −1 , cN ) be a sequence of real2N
√
2N
that
2 = cN < cN −1 < . . . < c1 < c0 = 2 and ci >
2ci+1 , for all
i ∈ {0, 1, . . . , N − 1}. Then, by the pigeonhole principle, for sufficiently
big n, there exists a subset Sn (1) ⊆ Sn such that |Sn (1)| ≥ cn1 and
|g1 W (x1 ) − g2 W (x1 )| < 2N√1 2n , ∀g1 , g2 ∈ Sn (1).
Suppose now 1 ≤ k ≤ N − 2, and Sn ⊇ Sn (1) ⊇ . . . ⊇ Sn (k) such
that for all j ∈ {1, . . . , k}, |Sn (j)| ≥ cnj and for all g1 , g2 ∈ Sn (j) we
have
1
√ n , 1 ≤ i ≤ j.
|g1 W (xi ) − g2 W (xi )| < 2N
2
Then by applying the pigeonhole principle to the set Sn (k) for sufficiently big n, we obtain Sn (k + 1) ⊆ Sn (k) such that |Sn (k + 1)| ≥ cnk+1 ,
and for all g1 , g2 ∈ Sn (k + 1) we have
1
√ n , 1 ≤ i ≤ k + 1.
|g1 W (xi ) − g2 W (xi )| < 2N
2
Then, for k = N −2, we obtain the desired inequality (condition (i)).
Now, let
h1 = g1 W, h2 = g2 W, yi = W (xi ), zi0 = g1 (yi ), zi00 = g2 (yi ), 1 ≤ i ≤ N.
Without loss of generality, we may also assume that g2 (y1 ) ≥ g1 (y1 ).
Then for all i ∈ {1, . . . , N − 1}, we have
−1
|h−1
1 h2 (xi ) − xi | = |(g1 W ) (g2 W )(xi ) − xi | =
|(g1 W )−1 (g2 W )(xi )−(g1 W )−1 (g1 W )(xi )| = |W −1 g1−1 g2 (yi )−W −1 g1−1 g1 (yi )|
= |W −1 g1−1 (zi00 ) − W −1 g1−1 (zi0 )|
Let u be a prefix of the reduced word g1 , and g1 = uv (so a reduced
word v is a suffix of g1 ). Then, since g1 , g2 ∈ Sn , we have
u−1 (zi0 ) = v(yi ) ≥ v(y1 ) ≥ y1
9
and
u−1 (zi00 ) = u−1 (g2 (yi )) ≥ u−1 (g2 (y1 )) ≥ u−1 (g1 (y1 )) ≥ v(y1 ) ≥ y1
Then by the Mean Value Theorem, we have
1
m
n 0
00
m
n
√ n
|h−1
1 h2 (xi ) − xi | ≤ M (θN ) |z1 − z1 | < M (θN ) 2N
2
Then, by condition (ii), we obtain |h−1
1 h2 (xi ) − xi | < . Then we have
−1
|h1 h2 (x) − x| < 2 for all x ∈ [0, 1]. Indeed, let x ∈ [xi , xi+1 ]. Then
−1
−1
|h−1
1 h2 (x) − x| ≤ max{|h1 h2 (xi ) − x|, |h1 h2 (xi+1 ) − x|}
−1
But |h−1
1 h2 (xi ) − x| ≤ |h1 h2 (xi ) − xi | + |xi − x| <
−1
−1
|h1 h2 (xi+1 ) − x| ≤ |h1 h2 (xi+1 ) − xi+1 | + |xi+1 − x|
2, and similarly,
< 2.
Since is arbitrary, we obtain that Γ is not C0 -discrete. By examining the proof of Proposition 1, we will now prove Theorem
A thus obtaining a much stronger result. The inequality |Sn | ≥ 2n is a
crucial fact in the proof of Proposition 1; we need the cardinality of Sn
grow exponentially. If Γ is an arbitrary finitely generated group with
exponential growth, this exponential growth of Sn is not automatically
guaranteed. But we can replace Sn by another subset Sn which still
does the job of Sn and which grows exponentially, if we assume a mild
condition on Γ.
First we need the following easy
Lemma 1. Let α, β ∈ G, z0 ∈ (0, 1) such that z0 ≤ α(z0 ) ≤ βα(z0 ).
Then U βα(z0 ) ≥ z0 where U = U (α, β) is any positive word in letters
α, β. Now we are ready to prove Theorem A.
Proof. Without loss of generality, we may assume that Γ has no
fixed point on (0, 1). Let again > 0, N ∈ N, δ > 0, θN >√0, M =
2sup0≤x≤1 (|f 0 (x)| + |g 0 (x)|) such that 1/N < , 1 < θN < 2N 2, and
for all x ∈ [1 − δ, 1], the inequality θ1N < φ0 (x) < θN holds where
φ ∈ {f, g, f −1 , g −1 }.
Let W = W (f, g) be an element of Γ such that
({f i W (1/N ) | − 2 ≤ i ≤ 2} ∪ {g i W (1/N ) | − 2 ≤ i ≤ 2}) ⊂ [1 − δ, 1]
and let m be the length of the reduced word W . Let also xi = i/N, 0 ≤
i ≤ N and z = W (1/N ).
10
By replacing the pair (f, g) with (f −1 , g −1 ) if necessary, we may
assume that f (z) ≥ z. Then at least one of the following cases is valid:
Case 1. f (z) ≤ gf (z);
Case 2. z ≤ gf (z);
Case 3. gf (z) ≤ z.
If Case 1 holds then we let α = f, β = g, z0 = z. If Case 1 does
not hold but Case 2 holds, then we let α = gf, β = f, z0 = z. Finally,
if Case 1 and Case 2 do not hold but Case 3 holds, then we let α =
f −1 g −1 , β = g −1 , z0 = gf (z).
In all the three cases, we will have z0 ∈ [1 − δ, 1], z0 ≤ z, and α, β
generate a free subsemigroup, and conditions of Lemma 1 are satisfied, i.e. we have z0 ≤ α(z0 ) ≤ βα(z0 ). Moreover, we notice that
sup0≤x≤1 (|α0 (x)| + |β 0 (x)|) ≤ M 2 , and the length of W in the alphabet
{α, β, α−1 , β −1 } is at most 2m.
Now, for every n ∈ N, let
Sn = {U (α, β)βαW | U (α, β) is a positive word in α, β of length at most n.}
Applying Lemma 1 to the pair {α, β} we obtain that V W −1 (z0 ) ≥ z0
for all V ∈ Sn .
Then |Sn | ≥ 2n . After achieving this inequality, we proceed as in
the proof of Proposition 1 with just a slight change: there exists a
sufficiently big n such that the following two conditions are satisfied:
(i) there exist g1 , g2 ∈ Sn such that g1 6= g2 , and
1
√ n , 1 ≤ i ≤ N − 1.
|g1 W (xi ) − g2 W (xi )| < 2N
2
(ii) M 2m+4 (θN )n
1
√ n
2N
2
< .
Let
h1 = g1 W, h2 = g2 W, yi = W (xi ), zi0 = g1 (yi ), zi00 = g2 (yi ), 1 ≤ i ≤ N.
Without loss of generality, we may also assume that g2 (y1 ) ≥ g1 (y1 ).
Then for all i ∈ {1, . . . , N − 1}, we have
−1
|h−1
1 h2 (xi ) − xi | = |(g1 W ) (g2 W )(xi ) − xi | =
|(g1 W )−1 (g2 W )(xi )−(g1 W )−1 (g1 W )(xi )| = |W −1 g1−1 g2 (yi )−W −1 g1−1 g1 (yi )|
= |W −1 g1−1 (zi00 ) − W −1 g1−1 (zi0 )|
11
Since g1 , g2 ∈ Sn , by the Mean Value Theorem, we have
1
2m+4
√ n
(θN )n |z10 − z100 | < M 2m+4 (θN )n 2N
|h−1
1 h2 (xi ) − xi | ≤ M
2
By condition (ii), we obtain that |h−1
1 h2 (xi ) − xi | < . Then we have
|h−1
h
(x)
−
x|
<
2
for
all
x
∈
[0,
1].
Since is arbitrary, we obtain
2
1
that Γ is not C0 -discrete. Proof of Theorem B. Let H be an arbitrary finitely generated
subgroup of [Γ, Γ]. If H contains a non-abelian free subsemigroup then
we are done by Theorem A. If H does not contain a non-abelian free
subsemigroup then by the result from [N2] H is virtually nilpotent.
Then again by the result of [N2], H is solvable of solvability degree at
most l(c). Since the natural number l(c) depends only on c, and not on
H, and since H is an arbitrary finitely generated subgroup of [Γ, Γ] we
obtain that [Γ, Γ] is solvable of solvability degree at most l(c). Hence
Γ is solvable with a solvability degree at most l(c) + 1. Proof of Theorem C. Let again H be an arbitrary finitely generated subgroup of [Γ, Γ]. Again, if H contains a non-abelian free
subsemigroup then we are done by Theorem A. If H does not contain
a non-abelian free subsemigroup then by the result from [N2] H is
virtually nilpotent. Then, by the result of Plante-Thurston ([PT]),
H is virtually Abelian. Then, by the result of Szekeres ([S]), H is
Abelian. Since H is an arbitrary finitely generated subgroup of [Γ, Γ],
we conclude that [Γ, Γ] is Abelian, hence Γ is metaabelian. R e f e r e n c e s:
[A] Akhmedov, A. On free discrete subgroups of Diff(I). Algebraic
and Geometric Topology, vol.4, (2010) 2409-2418.
[FF] Farb, B., Franks, J. Groups of homeomorphisms of one-manifolds
III: Nilpotent subgroups. Ergodic Theory and Dynamical Systems 23
(2003), 1467-1484
[N1] Navas, A. Sur les rapprochements par conjugasion en dimension
1 et classe C 1 . http://arxiv.org/pdf/1208.4815.
[N2] Navas, A. Growth of groups and diffeomorphisms of the interval.
Geom. Funct. Anal. 18 2008, no.3, 988-1028.
12
[PT] Plante, J., Thurston, W. Polynomial growth in holonomy
groups of foliations. Comment. Math. Helv. 51 (1976), 567-584.
[R] Raghunathan,M.S. Discrete subgroups of semi-simple Lie groups.
Springer-Verlag, New York 1972. Ergebnisse der Mathematik und ihrer
Grenzgebiete, Band 68.
[S] Szekeres, G. Regular iteration of real and complex functions.
Acta Math. 100 (1958), 203-258
Azer Akhmedov, Department of Mathematics, North Dakota State
University, Fargo, ND, 58108, USA
E-mail address: [email protected]