**1**

vote

**0**answers

81 views

### Presentation of hyperbolic groups [on hold]

Is it true that all hyperbolic groups are finitely presented? If yes, what is the right reference for that?

**6**

votes

**2**answers

353 views

### Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...

**14**

votes

**3**answers

675 views

### Failure of Mostow rigidity in dimension 2

I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question:
(1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...

**13**

votes

**2**answers

803 views

### Dehn's algorithm for word problem for surface groups

For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...

**14**

votes

**1**answer

247 views

### Does the injection $\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ split?

Let $F_n$ be the free group on $n$ letters.
The question is as in the title: letting $i:\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ be the natural injection, does there exist a homomorphism ...

**12**

votes

**2**answers

297 views

### When are growth series rational?

For a finitely generated group $G$ with generating set $S$, one can define the word metric. Using this, we are able to define the sequence $\sigma(n)$ by
$ \sigma(n) = |\mathcal{S}(e,n)| $, which is ...

**7**

votes

**0**answers

106 views

### An infinite torsion group $G$ with finite type $K(G,1)$?

There is a famous open problem in group theory that asks:
Does there exist an infinite finitely presented torsion group?
The general belief being that such groups exist. I would like to know ...

**6**

votes

**2**answers

540 views

### Geometric or topological results from group theory

Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples:
...

**15**

votes

**0**answers

259 views

### Groups whose finite index subgroups of fixed index are isomorphic

I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...

**10**

votes

**2**answers

846 views

### Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...

**5**

votes

**0**answers

93 views

### Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of ...

**6**

votes

**0**answers

171 views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of ...

**4**

votes

**2**answers

181 views

### Non-Cayley expander graphs

When I search about expander graphs in google I see a lot of articles about expander Cayley graphs. Now my questions are as follows:
Are all expander regular graphs are Cayley, or there is a special ...

**5**

votes

**2**answers

117 views

### Immersed quasi-Fuchsian surfaces surviving Dehn fillings

In papers like, Cooper - Long - Some surface subgroups survive surgery or Li - Immersed essential surfaces in hyperbolic 3-manifolds the game is to find some quasi-Fuchsian immersed surface $Q ...

**3**

votes

**1**answer

127 views

### Torsion-free, normal subgroups of certain Coxeter groups

Let $G$ be the reflection group of a regular, 4-dimensional, hyperbolic honeycomb. I would like to find a family $H_i < G$ of finite-index, torsion-free subgroups of $G$, so that I can represent ...

**4**

votes

**0**answers

133 views

### Log-concavity of the growth function

Given a Cayley graph of a group $G$ with finite generating set $A$ and exponential growth. Let $S_n$ be the elements whose word length is exactly $n$.
$\textbf{Question:}$ Is $f(n) = |S_{2n}|$ a ...

**9**

votes

**2**answers

578 views

### Folner sets and balls

Several related questions were asked before on MO, but it is not clear to me if the following was settled.
Given a finitely generated amenable group, is it always possible to find some finite ...

**14**

votes

**1**answer

578 views

### Is it true that every f.g. infinite simple group has exponential growth?

Is it true that every finitely generated infinite simple group has
exponential (word-)growth?
Remark: As Mark Sapir has pointed out, the question whether
every finitely generated group of ...

**6**

votes

**1**answer

319 views

### Liouville property - a very basic question

Let $\mathbb{F}_2$ be the free group on two generators. By a result of Kaimanovich and Vershik, for each measure $\mu$ on $\mathbb{F}_2$ such that the support of $\mu$ generates $\mathbb{F}_2$, we ...

**4**

votes

**0**answers

127 views

### Groups with infinitely many finite conjugacy classes

I've been coming across the condition "IMFCC: having infinitely many finite conjugacy classes" often in recent times and I was wondering if there is any serious difference between having "IMFCC" and ...

**12**

votes

**2**answers

259 views

### Are finitely generated amenable groups positively finitely generated?

Let $G$ be a finitely generated amenable group.
Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability?
Being more formal, note that $G^n$ is ...

**2**

votes

**0**answers

68 views

### Question about martin boundaries of random walks induced on transient subgroups

Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
...

**1**

vote

**0**answers

99 views

### Extending continuous functions from $\partial X$ to $X\cup \partial X$

Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial ...

**4**

votes

**2**answers

325 views

### A question about generating set of groups and epimorphism

Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an ...

**11**

votes

**1**answer

332 views

### Isometries of some simple Cayley graphs

Consider a Cayley graph of a group $G$ with respect to a symmetric finite generating set $S$.
There are some obvious candidates to isometries of this graph - for example, translation by elements of ...

**5**

votes

**0**answers

100 views

### Mal'cev completions of finitely generated torsion-free nilpotent groups

There is some question from geometric group theory:
One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$:
$\Gamma$ and ...

**6**

votes

**1**answer

187 views

### Generators of pure braid groups of arbitrary Coxeter groups

Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...

**5**

votes

**2**answers

184 views

### comparing homology of a space and homology of the classifying space of its fundamental group

Let $X$ be a (connected) closed $n$-manifold and $G=\pi_1(X)$ be the fundamental group of $X$. There is a classifying map $f: X \rightarrow K(G, 1)$ which induces an isomorphism on $\pi_1$. I would ...

**8**

votes

**1**answer

196 views

### Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?

Let $G$ be a finite group of diffeomorphisms of the torus $T^n$ fixing some point $p$, i.e. $p$ is fixed by every element of $G$. I have two questions.
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) ...

**7**

votes

**1**answer

368 views

### Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4

I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...

**5**

votes

**1**answer

117 views

### boundary of semihyperbolic groups

There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)?
The example I have in mind is a semisimple ...

**9**

votes

**4**answers

578 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 ...

**0**

votes

**0**answers

45 views

**4**

votes

**0**answers

105 views

### Invertibility of group Laplacian in $\ell^1$

Let $G$ be a discrete group and let $S$ be a generating set for $G$; assume that $S$ is symmetric (i.e., $g\in S$ iff $g^{-1}\in S$). Let $L=L_S=\frac{1}{|S|}(\sum_{g\in S} g-1)$ be an element of the ...

**12**

votes

**1**answer

261 views

### Hyperbolic 3-manifold groups acting on the plane

Can the fundamental group of a closed hyperbolic 3-manifold act freely on the plane by homeomorphisms? Freely and cocompactly? Freely, cocompactly, and preserving orientation?

**8**

votes

**2**answers

222 views

### Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...

**2**

votes

**0**answers

167 views

### Extra large spherical joins

If $X$ and $Y$ are piecewise spherical complexes, then their spherical join $X * Y$ is CAT(1) if and only of $X$ and $Y$ are CAT(1) (see the appendix of the first Charney-Davis paper below). One of ...

**3**

votes

**0**answers

146 views

### Are all finitely generated subgroups of SL2 LERF? [closed]

Let $H \leq \mathrm{SL}_2(\mathbb{R})$ be a finitely generated
subgroup. Must $H$ be LERF?
A group $H$ is said to be LERF (locally extended residually finite), or subgroup separable, if its ...

**7**

votes

**1**answer

166 views

### Subgroups of the mapping class group of a surface generated by Dehn twists

Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a ...

**10**

votes

**1**answer

458 views

### What is the Status of Borel conjecture today?

Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.

**3**

votes

**2**answers

615 views

### hyperbolic amenable graph

Is there an infinite (finite degree) transitive amenable hyperbolic graph ?

**0**

votes

**0**answers

63 views

### Joint point of coarse geometry and dynamical system?

My major interest is on dynamical systems,
but I did REU in a coarse embedding problem.
I wonder whether there's some significant connection between those two subjects.
I've tried to google for a ...

**11**

votes

**1**answer

268 views

### Growth of Poincaré duality groups

Can one prove that Poincaré duality groups cannot have intermediate growth?

**0**

votes

**0**answers

107 views

### Poincaré inequality for connected Lie groups

Let $G$ be a compactly generated second countable locally compact group, and let $\mu$ be a probability measure which is:
symmetric,
adapted (in the sense that there is no proper subgroup $H$ such ...

**8**

votes

**1**answer

165 views

### Does every generating set of the first homology group of a Cayley graph give rise to a presentation of its group?

Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph.
Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set ...

**0**

votes

**0**answers

69 views

### Rigidity of lower-dimensional lattices in Euclidean groups

Informal intro / motivation:
Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer linear ...

**3**

votes

**0**answers

114 views

### Short exact sequences for amalgamated free products and HNN Extensions

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too:
If $A$ and $B$ are groups we have the following short exact sequence:
$$ 0 \to ...

**3**

votes

**2**answers

328 views

### Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated?
In view of Ian Agol's answer, I ...

**1**

vote

**0**answers

53 views

### Fully residually free groups and completion

Let $G$ be a fully residually free group with a finitely generated profinite completion. Is $G$ necessarily finitely generated?

**5**

votes

**0**answers

170 views

### Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) ...