Geometric and asymptotic group theory
Olga Kharlampovich
(McGill University)
Winter School, 2010
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i,
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
That is groups generated by a1 , ..., an
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
That is groups generated by a1 , ..., an with relations
r1 = 1, r2 = 1, ... imposed.
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
That is groups generated by a1 , ..., an with relations
r1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with two
generators is given by the presentation
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
That is groups generated by a1 , ..., an with relations
r1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with two
generators is given by the presentation
ha, b | u n = 1i
for all words u in the alphabet a, b.
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
That is groups generated by a1 , ..., an with relations
r1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with two
generators is given by the presentation
ha, b | u n = 1i
for all words u in the alphabet a, b.
The fundamental group of the orientable surface of genus n is
given by the presentation
Group presentations
The object of our study - finitely generated groups given by
presentations ha1 , ..., an | r1 , r2 , ...i, where ri is a word in a1 , ..., an .
That is groups generated by a1 , ..., an with relations
r1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with two
generators is given by the presentation
ha, b | u n = 1i
for all words u in the alphabet a, b.
The fundamental group of the orientable surface of genus n is
given by the presentation
ha1 , b1 , ..., an , bn | [a1 , b1 ]...[an , bn ] = 1i.
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem)
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Theorem. (Higman)
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has recursively enumerable word
problem iff
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has recursively enumerable word
problem iff it is a subgroup of a finitely presented group.
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has recursively enumerable word
problem iff it is a subgroup of a finitely presented group.
Theorem. (Adian-Novikov’s solution of Burnside problem)
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has recursively enumerable word
problem iff it is a subgroup of a finitely presented group.
Theorem. (Adian-Novikov’s solution of Burnside problem) The
free Burnside group of exponent n with at least two generators
Some classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) There
exists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has recursively enumerable word
problem iff it is a subgroup of a finitely presented group.
Theorem. (Adian-Novikov’s solution of Burnside problem) The
free Burnside group of exponent n with at least two generators is
infinite for large enough odd n.
Some classical results
The growth rate of a group is a well-defined notion from
asymptotic analysis. To say that a finitely generated group has
polynomial growth means the number of elements of length
(relative to a symmetric generating set) at most n is bounded
above by a polynomial function p(n). The order of growth is then
the least degree of any such polynomial function p.
Some classical results
The growth rate of a group is a well-defined notion from
asymptotic analysis. To say that a finitely generated group has
polynomial growth means the number of elements of length
(relative to a symmetric generating set) at most n is bounded
above by a polynomial function p(n). The order of growth is then
the least degree of any such polynomial function p.
A nilpotent group G is a group with a lower central series
terminating in the identity subgroup.
Some classical results
The growth rate of a group is a well-defined notion from
asymptotic analysis. To say that a finitely generated group has
polynomial growth means the number of elements of length
(relative to a symmetric generating set) at most n is bounded
above by a polynomial function p(n). The order of growth is then
the least degree of any such polynomial function p.
A nilpotent group G is a group with a lower central series
terminating in the identity subgroup.
Theorem. (Gromov’s solution of Milnor’s problem)
Some classical results
The growth rate of a group is a well-defined notion from
asymptotic analysis. To say that a finitely generated group has
polynomial growth means the number of elements of length
(relative to a symmetric generating set) at most n is bounded
above by a polynomial function p(n). The order of growth is then
the least degree of any such polynomial function p.
A nilpotent group G is a group with a lower central series
terminating in the identity subgroup.
Theorem. (Gromov’s solution of Milnor’s problem) Any group of
polynomial growth
Some classical results
The growth rate of a group is a well-defined notion from
asymptotic analysis. To say that a finitely generated group has
polynomial growth means the number of elements of length
(relative to a symmetric generating set) at most n is bounded
above by a polynomial function p(n). The order of growth is then
the least degree of any such polynomial function p.
A nilpotent group G is a group with a lower central series
terminating in the identity subgroup.
Theorem. (Gromov’s solution of Milnor’s problem) Any group of
polynomial growth has a nilpotent subgroup of finite index.
Finitely generated groups viewed as metric spaces
Let G be a group given as a quotient π : F (S) → G of the free
group on a set S. Therefore G = hS|Ri. The word length |g | of an
element g ∈ G is the smallest integer n for which there exists a
sequence s1 , . . . , sn of elements in S ∪ S −1 such that
g = π(s1 . . . sn ). The word metric dS (g1 , g2 ) is defined on G by
dS (g1 , g2 ) = |g1−1 g2 |.
G acts on itself from the left by isometries.
Definition Let π : F (S) → G as above. The corresponding Cayley
Graph Cay (G , S) is the graph with vertex set G in which two
vertices g1 , g2 are the ends of an edge if and only if dS (g1 , g2 ) = 1.
Cayley graph of the free group F (a, b)
Cayley graph of the Dihedral group
D4 = hα, β|α4 , β 2 , αβαβi
Cayley graph
Note, that if S and S̄ are two finite generating sets of G then dS
and dS̄ are bi-Lipschitz equivalent.
Polynomial growth
A ball of radius n in Cay (G , S) is
Bn = {g ∈ G ||g | ≤ n}.
G has polynomial growth iff the number of elements in Bn is
bounded by a polynomial p(n).
Hyperbolic groups
A geodesic metric space is called δ-hyperbolic if for every geodesic
triangle, each edge is contained in the δ neighborhood of the union
of the other two edges. If δ = 0 the space is called a real tree or
R-tree.
A group G is hyperbolic Cay (G , X ) is hyperbolic.
Quasi-isometry
Definition Let (X , dX ) and (Y , dY ) be metric spaces. Given real
numbers k ≥ 1 and C ≥ 0, a map f : X → Y is called a
(k, C )-quasi-isometry if
1.
1
k dX (x1 , x2 )
x1 , x2 ∈ X ,
− C ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ) + C for all
2. the C neighborhood of f (X ) is all of Y .
Examples of quasi-isometries
1. (Z; d) and (R; d) are quasi-isometric.
The natural embedding of Z in R is isometry. It is not surjective,
but each point of R is at most 1/2 away from Z.
2. All regular trees are quasi-isometric
All regular trees of valence at least 3 are quasi-isometric. We
denote by Tk the regular tree of valence k and we show that T3 is
quasi-isometric to Tk for every k ≥ 4. We define the map
q : T3 → Tk , sending all edges drawn in thin lines isometrically
onto edges and all paths of length k − 3 drawn in thick lines onto
one vertex. The map q thus defined is surjective and it satisfies
the inequality
1
dist(x, y ) − 1 ≤ dist(q(x), q(y )) ≤ dist(x, y ).
k −2
3. Free groups of finite rank are quasi-isometric
All non-Abelian free groups of finite rank are quasi-isometric to
each other. The Cayley graph of the free group of rank n with
respect to a set of n generators and their inverses is the regular
simplicial tree of valence 2n.
4. Let G be a group with a finite generating set S, and let
Cay (G , S) be the corresponding Cayley graph. We can make
Cay (G , X ) into a metric space by identifying each edge with a unit
interval [0, 1] in R and defining d(x, y ) to be the length of the
shortest path joining x to y . This coincides with the path-length
metric when x and y are vertices. Since every point of Cay (G , X )
is in the 1/2 -neighbourhood of some vertex, we see that G and
Cay (G , S) are quasi-isometric for this choice of d.
Quasi-isometry
5. Every bounded metric space is quasi-isometric to a point.
6. If S and T are finite generating sets for a group G , then (G , dS )
and (G , dT ) are quasi-isometric.
7. The main example, which partly justifies the interest in
quasi-isometries, is the following. Given M a compact Riemannian
manifold, let M̃ be its universal covering and let π1 (M) be its
fundamental group. The group π1 (M) is finitely generated, in fact
even finitely presented.
The metric space M̃ with the Riemannian metric is quasi-isometric
to π1 (M) with some word metric.
8. If G1 is a finite index subgroup of G , then G and G1 are
quasi-isometrically equivalent.
Quasi-isometric rigidity
Classes of groups complete with respect to quasi-isometries
I
Finitely presented groups,
I
Nilpotent groups,
I
Abelian groups,
I
Hyperbolic groups,
I
nonabelian free groups of finite rank (follows from the fact
that their Caley graphs are trees).
I
Amenable groups (see below)
Quasi-isometric rigidity
Eskin, Fisher, Whyte obtained first results on quasi-isometric
rigidity of non-nilpotent polycyclic groups.
Theorem Any group quasi-isometric to the three dimensional
solvable Lie group Sol is virtually a lattice in Sol.
That completed the classification of three-dimensional manifolds
up to quasi-isometry started by Thurston, Schwartz and others.
Conjecture Let G be a solvable Lie group, and Γ a lattice in G .
Any finitely generated group Γ0 quasi-isometric to Γ is virtually a
lattice in a solvable Lie group G 0 .
Equivalently, any f.g. group quasi-isometric to a polycyclic group is
virtually polycyclic.
Limit groups (fully residually free groups)
A marked group (G , S) is a group G with a prescribed family of
generators S = (s1 , . . . , sn ).
Two marked groups (G , (s1 , . . . , sn )) and (G 0 , (s10 , . . . , sn0 )) are
isomorphic as marked groups if the bijection si ←→ si0 extends to
an isomorphism. For example, (hai, (1, a)) and (hai, (a, 1)) are not
isomorphic as marked groups. Denote by Gn the set of groups
marked by n elements up to isomorphism of marked groups.
One can define a metric on Gn by setting the distance between two
marked groups (G , S) and (G 0 , S 0 ) to be e −N if they have exactly
the same relations of length at most N (under the bijection
S ←→ S 0 ) (Grigorchuk, Gromov’s metric)
Finally, a limit group is a limit (with respect to the metric above)
of marked free groups in Gn .
limits of free groups
Example: A free abelian group of rank 2 is a limit of a sequence of
cyclic groups with marking
(hai, (a, an )), n → ∞.
limits of free groups
Theorem Let G be a finitely generated group. Then the following
conditions are equivalent:
1) G is freely discriminated (that is for finitely many non-trivial
elements g1 , . . . , gn ∈ G there exists a homomorphism φ from
G to a free group such that φ(gi ) 6= 1 for i = 1, . . . , n);
2) [Remeslennikov] G is universally equivalent to F (in the
language without constants);
3) [Baumslag, Kharlampovich, Myasnikov, Remeslennikov] G is
the coordinate group of an irreducible variety over a free
group.
4) [Champetier and Guirardel] G is a limit of free groups in
Gromov-Grigorchuk metric.
Problem from logic solved using asymptotic methods
Theorem [Kharlampovich, Myasnikov, independently Sela
2006]
The elementary theories of free non-abelian groups Fn and Fm
coincide.
This was Alfred Tarski’s problem from 1945.
Recall that the elementary theory Th(G ) of a group G is the set of
all first order sentences in the language of group theory which are
true in G .
Example os a sentence in the theory of F :
∀x∀y ∃z([x, y ] = 1 → (x = z 2 ∨ y = z 2 ∨ xy = z 2 ))
This implies that if a group G is ∀∃ equivalent to F , then it does
not have non-cyclic abelian subgroups.
Free actions on metric spaces
A group G acts on a set X if for each g ∈ G there is a bijection
x → gx defined on X such that
ex = x, (g1 (g2 (x)) = (g1 g2 )(x).
Theorem. A group G is free if and only if it acts freely by
isometries on a tree.
Free action = no inversion of edges and stabilizers of vertices are
trivial.
R-trees
An R-tree is a metric space (X , p) (where p : X × X → R) which
satisfies the following properties:
1) (X , p) is geodesic,
2) if two segments of (X , p) intersect in a single point, which is
an endpoint of both, then their union is a segment,
3) the intersection of two segments with a common endpoint is
also a segment.
R-trees
An R-tree is a metric space (X , p) (where p : X × X → R) which
satisfies the following properties:
1) (X , p) is geodesic,
2) if two segments of (X , p) intersect in a single point, which is
an endpoint of both, then their union is a segment,
3) the intersection of two segments with a common endpoint is
also a segment.
Examples
X = R with usual metric.
Examples
X = R with usual metric.
A geometric realization of a simplicial tree.
Examples
X = R with usual metric.
A geometric realization of a simplicial tree.
X = R2 with metric d defined by
|y1 | + |y2 | + |x1 − x2 | if x1 6= x2
d((x1 , y1 ), (x2 , y2 )) =
|y1 − y2 |
if x1 = x2
(x1,y1)
x
(x2,y2)
X = R2 with SNCF metric (French Railway System)
Finitely generated R-free groups
Rips’ Theorem [Rips, 1991 - not published]
A f.g. group is R-free (acts freely on an R-tree) if and only if it is
a free product of surface groups (except for the non-orientable
surfaces of genus 1,2, 3) and free abelian groups of finite rank.
Finitely generated R-free groups
Rips’ Theorem [Rips, 1991 - not published]
A f.g. group is R-free (acts freely on an R-tree) if and only if it is
a free product of surface groups (except for the non-orientable
surfaces of genus 1,2, 3) and free abelian groups of finite rank.
Gaboriau, Levitt, Paulin (1994) gave a complete proof of Rips’
Theorem.
Finitely generated R-free groups
Rips’ Theorem [Rips, 1991 - not published]
A f.g. group is R-free (acts freely on an R-tree) if and only if it is
a free product of surface groups (except for the non-orientable
surfaces of genus 1,2, 3) and free abelian groups of finite rank.
Gaboriau, Levitt, Paulin (1994) gave a complete proof of Rips’
Theorem.
Bestvina, Feighn (1995) gave another proof of Rips’ Theorem
proving a more general result for stable actions on R-trees.
Ordered abelian groups
Λ = an ordered abelian group.
Ordered abelian groups
Λ = an ordered abelian group.
Examples:
Archimedean case:
Λ = R, Λ = Z with the usual order.
Ordered abelian groups
Λ = an ordered abelian group.
Examples:
Archimedean case:
Λ = R, Λ = Z with the usual order.
Non-Archimedean case:
Λ = Z2 with the left lexicographic order:
(a, b) < (c, d) ⇐⇒ b < d or b = d and a < c.
Z2 with the left-lex ordering
y
(0,1)
(0,0)
(0,-1)
x
Z2 with the left-lex ordering
y
(0,1)
x
(0,0)
(0,-1)
One-dimensional picture
((
(0,-1)
((
(0,0)
((
(0,1)
((
Λ-trees
Morgan and Shalen (1985), Bass (1987) defined Λ-trees:
A Λ-tree is a Λ-metric space enjoying the properties listed in the
definition of R-trees with R substituted by Λ.
Λ-trees
Morgan and Shalen (1985), Bass (1987) defined Λ-trees:
A Λ-tree is a Λ-metric space enjoying the properties listed in the
definition of R-trees with R substituted by Λ.
Find the group theoretic information carried by a free action on a
Λ-tree.
Conjecture (Kharlampovich, Miasnikov) Every f.g. Λ-free group
is Zn -free for suitable n. (We described f.g. Zn -free groups.)
Asymptotic cones
GromovHausdorff distance measures how far two compact metric
spaces are from being isometric. If X and Y are two compact
metric spaces, then dGH (X , Y ) is defined to be the infimum of all
numbers dH (f (X ), g (Y )) for all metric spaces M and all isometric
embeddings f : X → M and g : Y → M.
Definition (Gromov, van den Dries - Wilkie) An asymptotic
cone of a group G corresponding to a sequence of scalars dn → ∞
is a Gromov-Hausdorff limit of the spaces G /dn , where G /dn is the
Cayley graph of G with normalized metric (distance f-n is divided
by dn ).
Examples 1. A f.g. group is virtually abelian iff its a.c. are
isometric to the Euclidean space Rn .
2. A f.g. group is virtually nilpotent iff its a. c. are locally
compact.
3. A f.g. group is hyperbolic iff its a.c. are R-trees.
The power of asymptotic cones stems from the fact that they
capture both geometrical and logical properties of the group.
Asymptotic cones
Theorem (Gromov) If all a.c. of a group are simply connected
then the group has polynomial Dehn function and linear
isodiametric function.
Papasoglu: The converse statement is true if the Dehn function is
quadratic.
Theorem (Olshanskii, Sapir) There are f.p. groups with
non-simply connected asymptotic cones but Dehn functions
arbitrary close to n2 and linear isodiametric function.
Theorem (Olshanskii, Sapir) There are f.p. groups with
undecidable w.p. but Dehn function f (n) bounded by Cn2 for
infinitely many n0 s. Such a group has at least two
non-homeomorphic a.c.: one simply connected and one non-simply
connected.
Asymptotic cones
The notion of an asymptotic cone plays an important role in
Geometric group theory since asymptotic cones (or, more precisely,
their topological types and bi-Lipschitz types) provide
quasi-isometry invariants of metric spaces in general and of finitely
generated groups in particular.