**3**

votes

**1**answer

128 views

### Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups

Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence".
Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...

**2**

votes

**0**answers

87 views

### example of fuchsian groups acting on 2-sphere by G. Martin

Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (...

**1**

vote

**1**answer

118 views

### Subsets of the boundary of a surface group

Consider the surface group $\Gamma=\langle a,b,c,d\mid [a,b][c,d]=1\rangle$: it is a Gromov hyperbolic group; its Gromov boundary $\partial\Gamma$ is homeomorphic to $S^1$ (the unit circle).
I would ...

**6**

votes

**0**answers

50 views

### Coarse embeddability into Hilbert space of residually finite groups

By definition a finitely generated group G is coarsely embedded into Hilbert space if there is a function $F: G\to \ell_2$, such that $\|F(g_n)-F(h_n)\|\to\infty$ iff $d(g_n,h_n)\to\infty$, where $d$ ...

**12**

votes

**1**answer

136 views

### The finiteness criterium $F$ under quasi-isometry

A group $G$ is defined to have $F$ if there exists a finite $K(G,1)$.
This property is clearly not invariant under quasi-isometry as one can see from the trivial group and $\mathbb{Z}_2$.
My question:...

**5**

votes

**1**answer

167 views

### CAT(0)-groups in dimension 2

Suppose I have a space $X$ which is connected, simply connected, CAT(0) of dimension 2 and a group $G$ which acts on $X$ freely, isometrically, properly discontinuously and cocompactly. What can be ...

**1**

vote

**0**answers

110 views

### Is there a non-integer in the dimension spectrum for the Heisenberg group?

Let $\Gamma = \langle a,b,c \ | \ c=aba^{-1}b^{-1}, \ ac=ca, \ bc = cb \rangle$ be the discrete Heisenberg group.
Let $\ell: \Gamma \to \mathbb{N} $ be the word length on $\Gamma$. This group has a ...

**1**

vote

**0**answers

97 views

### Presentation of hyperbolic groups [closed]

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

**6**

votes

**2**answers

445 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 ...

**15**

votes

**1**answer

274 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 $...

**7**

votes

**0**answers

115 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 ...

**15**

votes

**0**answers

272 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 ...

**7**

votes

**2**answers

576 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:
...

**12**

votes

**2**answers

319 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 ...

**5**

votes

**0**answers

104 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 non-...

**6**

votes

**0**answers

188 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

190 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 ...

**3**

votes

**1**answer

129 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 ...

**6**

votes

**1**answer

330 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

132 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 ...

**2**

votes

**0**answers

72 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

107 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 X\to\...

**5**

votes

**0**answers

111 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 $\...

**12**

votes

**2**answers

265 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 ...

**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}) =...

**5**

votes

**2**answers

188 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 ...

**6**

votes

**1**answer

129 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 ...

**0**

votes

**0**answers

46 views

**4**

votes

**0**answers

117 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

263 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?

**2**

votes

**0**answers

175 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

150 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

179 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 ...

**8**

votes

**2**answers

232 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 ...

**10**

votes

**1**answer

469 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.

**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 ...

**7**

votes

**1**answer

197 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 ...

**4**

votes

**2**answers

328 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 ...

**8**

votes

**1**answer

167 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

123 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 [...

**1**

vote

**0**answers

56 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

173 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) \...

**1**

vote

**0**answers

82 views

### Profinite rank of Fuchsian groups

Let $G$ be a Fuchsian group whose profinite completion is finitely generated. Must $G$ be finitely generated?
A Fuchsian group is a discrete subgroup of $\mathrm{SL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(...

**3**

votes

**2**answers

333 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 ...

**4**

votes

**1**answer

184 views

### An inequality for Fuchsian groups?

Let $G$ be a finitely generated Fuchsian group.
(i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$).
Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ?
Here, $\beta_{2}^1(G)$ stands for ...

**2**

votes

**0**answers

119 views

### Is a finitely generated residually free group “almost LERF”?

Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
Let ...

**2**

votes

**0**answers

80 views

### Rank gradient in free products amalgamating a finite subgroup

Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...

**4**

votes

**1**answer

152 views

### Which 3-manifolds have positive rank gradient?

For which $3$-manifolds $M$ is the fundamental group $\pi_1(M)$
finitely generated and has positive rank gradient?
Recall that the rank gradient of a finitely generated group $G$ is defined to be $...

**5**

votes

**0**answers

124 views

### Volume growth of balls

Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. Suppose that on the homogeneous space $X=G/K$ we have a $G$-invariant proper metric $d$. For $R>0$ let $B(R)$ be the open ...