Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

learn more… | top users | synonyms

2
votes
0answers
107 views

Local curvature in a Cayley complex

I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local ...
10
votes
6answers
850 views

Understanding groups that are not linear.

I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely: What are some interesting ...
5
votes
1answer
316 views

Does $C'\left(\frac{5}{11}\right)$ imply exponential growth?

I came across this rather week small cancellation condition $C'\left(\frac{5}{11}\right)$ of a group $G$. It has been proved that $C'\left(\frac16\right)$ is enough for $G$ to contain free subgroups. ...
6
votes
2answers
220 views

Dehn function for undistorted subgroups of a product of free groups

Let $G$ be a finitely generated subgroup of a product of two finite rank free groups $F_m \times F_n$. If there is a Lipschitz retraction $F_m \times F_n \to G$ with respect to word metrics, then $G$ ...
2
votes
1answer
182 views

Lowering metrics to finite index subgroups

Let $G$ be an infinite, countable, finitely generated group. Let $H$ be a finite index subgroup of $G$. Let $S$ be a finite, symmetric set of generators of $G$, and let $d(\cdot,\cdot)$ be the word ...
1
vote
2answers
310 views

Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups? I know that ...
5
votes
0answers
199 views

Any method to detect subgroup generated by a subset of the generators from its presentation

I have met the following problem. A group $G$ is given as follows $G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$ Is the subgroup generated by $y$ and $t$ just the ...
2
votes
1answer
527 views

Fundamental Lemma Of Geometric Group Theory

I'll be delighted to get some help in understanding the proof of the first theorem here: http://www.math.utah.edu/~malone/QI/notes.pdf "If G acts geometrically on X and Y (proper geodesic metric ...
3
votes
1answer
533 views

French resources for (Geometric) Group Theory

I am looking for ways to improve my mathematical French while learning more material about either finite group theory or geometric group theory. In particular, I would love to find a French equivalent ...
20
votes
2answers
795 views

Asymptotics of the growth rate of a group

Let $\Gamma$ be a finitely generated group of exponential growth and $gr(S)=\lim_{k\rightarrow \infty} \sqrt[k]{|B_k(S)|}$ be the growth rate of $\Gamma$ with respect to the generating set $S$. I am ...
1
vote
2answers
162 views

Conjugated elements in amalgameted Product

Hello! Let $G=A\underset{C}\star B$ be an amalgamated Product. Let $a\in A$. If a is conjugated to an Element $b\in B$, then $a$ is conjugated to an Element $c\in C$. The Question is: Why is that ...
11
votes
2answers
472 views

Does every group grow either polynomially or superpolynomially?

I am reading an introduction to growth of groups. The notions of polynomial and superpolynomial growth are introduced, as are exponential and subexponential growth. I can prove that the growth of a ...
3
votes
2answers
336 views

Free groups as quotients of hyperbolic groups

Given any infinite non-elementary hyperbolic group $G$, a theorem of Gromov asserts that there is a subgroup of $G$ isomorphic to a non-abelian free group on two generators. Is there a similar result ...
9
votes
3answers
417 views

Finite subgroups of relatively hyperbolic groups

It is well known that in a given $\delta$-hyperbolic group there are only finitely many conjugacy classes of finite subgroups. This is clearly false for relatively hyperbolic groups since we have no ...
6
votes
1answer
200 views

Asymptotics of the number of required Dehn relators in hyperbolic groups

If $G = \langle X | R \rangle$ is a $\delta$-hyperbolic group presentation, then Dehn's algorithm provides a linear time solution to the word problem, but the linear constant is horribly exponential ...
16
votes
1answer
702 views

Amenable groups of deficiency $1$

Let $G=\langle X;R\rangle$ be a finitely presented group. The rank of $G$ is defined to be the size of smallest generating set of $G$. The deficiency ${\rm def}(G)$ of $G$ is defined to be the maximum ...
9
votes
1answer
359 views

Kazhdan's property T for Kahler surfaces

Is it true that the fundamental groups of compact Kahler surfaces have property T if and only if it they are finite? I am having trouble finding counterexamples to this, but maybe that's just me...
6
votes
2answers
270 views

Groups quasi-isometric to reducible nonuniform lattices

It is known that a finitely group $G$ is quasi-isometric to a nonuniform irreducible lattice $\Lambda$ in a semisimple Lie group if and only if $G$ and $\Lambda$ are commensurable (see references in ...
4
votes
1answer
379 views

Dehn Twist in the sense of Geometric Group Theory and a Graph of Groups

Hello! Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or a subgroup of a group. I ...
3
votes
1answer
329 views

Relationship between hyperbolicity in group theory and hyperbolicity in geometry

Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional ...
16
votes
1answer
434 views

Is the property of not containing $\mathbb{F}_2$ invariant under quasi-isometry?

Is the property of not containing the free group on two generators invariant under quasi-isometry? Amenability is, so if there is a counterexample it is also a solution to the von Neumann-Day ...
5
votes
1answer
236 views

Flat sector in a proper cocompact CAT(0) space

Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 ...
10
votes
3answers
1k views

Examples of “Monster” groups

I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are: 1.) ...
6
votes
5answers
723 views

Analogues of the dihedral group

A virtually-$\mathbb{Z}$ group $G$ admits either a epimorphism onto $\mathbb{Z}$ or a epimorphism onto $D_\infty$. So what happens if one replaces $\mathbb{Z}$ by another group $F$ (like the free ...
15
votes
3answers
945 views

The second homotopy group of a simple CW-complex

Let $X$ be a CW-complex with one 0-cell two 1-cells three 2-cells no cells in dimensions 3 or higher. Is it always true that $\pi_2(X)\ne 1$?
4
votes
1answer
295 views

Examples of CAT(0)-groups

My question is the following: Let M be a simply connected Riemannian manifold whose sectional curvatures are all nonpositive and let G be a group. Suppose that G acts in M properly discontinuous and ...
10
votes
2answers
430 views

Centralizers of non-iwip elements of $Out(F_n)$

Does there exist an infinite order element $\phi\in Out(F_n)$, for some or all $n\geq 3$, which is not iwip but has finite index in its centralizer? How about an element such that all its non-zero ...
6
votes
3answers
398 views

Are subgroups of hyperbolic groups quasiisometrically embedded ?

Given a finitely generated subgroup of a finitely generated hyperbolic group. Is it true that the inclusion of each subgroup is a quasiisometric embedding ? The first example for a group that does ...
4
votes
1answer
230 views

Asymptotic dimension of graph manifold groups

Does every non-geometric graph manifold have fundamental group of asymptotic dimension 3? This is affirmed in http://arxiv.org/abs/0909.1098 for closed graph manifolds, but I am interested in ...
8
votes
0answers
284 views

is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
5
votes
1answer
307 views

Growth of groups versus Schreier graphs

This question is motivated by this one What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes? where it essentially asks to compare the growth ...
0
votes
1answer
262 views

Hyperbolicity of a fundamental group

Let G be the fundamental group of a compact 3-manifold which supports on its interior a complete non positively curved Riemannian metric and is a cilinder near de metric. Is G hyperbolic?
4
votes
1answer
220 views

Hyperbolicity of a semidirect product

Let F be a finitely generated free group and let $\gamma : F \rightarrow F$ be an automorphism. Is the semidirect product $F \rtimes \mathbb{Z}$ an hyperbolic group? where $\mathbb{Z}$ acts in F via ...
18
votes
3answers
721 views

Is there a non-Hopfian lacunary hyperbolic group?

The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...
3
votes
1answer
354 views

Finite quotients of graphs

If $X$ is an infinite graph, $G$ is a group acting on $X$ with finite quotient; make $Y=X/G$ into graph of groups by attaching stabilizers at vertices and edges. Let $Z$ be a graph of groups, with ...
4
votes
1answer
398 views

Groups acting on graph

Let $S$ be an infinite graph, $G$ is a group acting (effectively) on $S$ with finite quotient graph $S/G$. Make $S/G$ into graph of groups in obvious way by assigning stabilizers at vertices and ...
8
votes
1answer
580 views

Surface groups and non separating loops

QUESTION: Let $g \geq 4$, $S(g)$ be the fundamental group of the genus $g$ surface, and $G$ be finitely generated (the number of generators $\leq 3$) group with abelianization of rank less than equal ...
1
vote
3answers
472 views

Serre for complexes of groups

A theorem of Serre states that any finite subgroup $F$ of an (injective) amalgam $G_1 *_H G_2$ is conjugate into one of the factors $G_1$ or $G_2$, that is, $F$ is a subgroup of a vertex stabilizer of ...
7
votes
3answers
385 views

Embedding groups into groups with some vanishing homology groups

Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H_i(\Gamma;\mathbb{Z})=0$ for all $i \in ...
5
votes
2answers
630 views

Residual Finiteness of Fundamental Group of Compact 3-Manifold

Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi_1(M)$ is residually finite. The outline of the proof is basically: ...
6
votes
3answers
496 views

Braid group analogue for signed symmetric group?

This is probably something well known (either in the affirmative or in the negative) but I couldn't get this information easily: Braid group:Symmetric group::?:Signed symmetric group By "signed ...
22
votes
4answers
1k views

Trees in groups of exponential growth

Question: Let $G$ be a finitely generated group with exponential growth. Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree? ...
5
votes
1answer
445 views

Percolation in Cayley graphs of semigroups.

Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...
5
votes
2answers
675 views

Quasi-isometries vs Cayley Graphs

The following questions might be trivial, however, I couldn't solve them: Let $G$ be generated by a finite symmetric set $S.$ Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of ...
16
votes
4answers
719 views

When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?

The question has an easy answer, if one replaces free by free abelian: Then the resulting group is always solvable and a solvable subgroup of a CAT(0) group is virtually abelian. If the resulting was ...
5
votes
2answers
393 views

Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?

Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely ...
10
votes
4answers
850 views

What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the most general setting for which it might expected to be true?

What I would like to know is exactly what the title asks: What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the ...
24
votes
2answers
1k views

Invertible matrices satisfying $[x,y,y]=x$.

I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope. Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and ...
14
votes
1answer
518 views

Loop spaces and infinite braids

The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$. The infinite symmetric group has interesting interactions with homotopy ...
2
votes
1answer
318 views

balls as Foelner sets

This is essentially equivalent to this question by Simon Thomas. Let $G=\langle X\rangle$ be a finitely generated group, $b_n$ be the number of elements in the ball of radius $n$ in the Cayley ...