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

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5
votes
0answers
63 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
0answers
128 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
2answers
173 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
1answer
125 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
1answer
315 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
0answers
125 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
0answers
64 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
0answers
97 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 ...
5
votes
0answers
90 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
2answers
255 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
1answer
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
2answers
180 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 ...
4
votes
1answer
110 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
0answers
45 views

Asymptotic dimension of Bicombable groups

Do Bicombable Groups have finite asymptotic dimension?
4
votes
0answers
98 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
1answer
259 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
0answers
162 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
0answers
145 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
1answer
161 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
2answers
218 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
1answer
457 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
0answers
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 ...
6
votes
1answer
185 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
2answers
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 ...
8
votes
1answer
160 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
0answers
68 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
0answers
113 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
0answers
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
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0answers
167 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
0answers
81 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 ...
3
votes
2answers
326 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 ...
3
votes
1answer
177 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 ...
2
votes
0answers
111 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 $\tau(g) \neq 1$.) ...
2
votes
0answers
77 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
1answer
146 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 ...
5
votes
0answers
113 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 ...
2
votes
1answer
128 views

Engulfing Kleinian groups?

Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ? I know that this is true for Fuchsian ...
6
votes
2answers
213 views

How bad is the modular space?

I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$? Do we know something about its homology or homotopy groups ? $\mathbb{H}^{3}$ is the hyperbolic ...
1
vote
0answers
66 views

Salvaging Howson's theorem for free profinite groups

This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem? Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
3
votes
2answers
233 views

Do limit groups satisfy Howson's theorem?

Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must $A \cap B$ be finitely generated? Recall that a limit ...
1
vote
0answers
88 views

Homology of spherical braid groups

By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
10
votes
2answers
298 views

Conjugacy of matrices of order three in $PGL(2,k)$, where $k$ is any field

We know that every matrix of order three in $PGL(2,\mathbb Z)$ is conjugate to the following matrix $$ \left( \begin{array}[cc] &1 & -1 \\ 1&0 \end{array} \right) $$ I want to know if ...
0
votes
0answers
120 views

Rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$

What is the rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$, where $n\geq 3$ ? A reference would be appreciated. For the definition of "cohomological dimension of a group ...
5
votes
0answers
265 views

Quotients of an extension of the Higman group

(Note: this started as a different question that soon changed form, thanks to the answers.) Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ ...
1
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0answers
72 views

Algorithm to generate hyperbolic metric on a compact surface

Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
5
votes
0answers
109 views

Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
5
votes
0answers
131 views

A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers

I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that a) the graph $\Gamma$ is fine, b) $\Gamma$ is not a tree, c) not all ...
3
votes
0answers
47 views

Cubulating non-compact hyperbolic manifolds

Let $X$ be a hyperbolic manifold of arbitrary dimension. When does $X$ admit a cell structure of a CAT(-1) cube complex? of a hyperbolic CAT(0) cube complex? I suspect that the question is widely ...
7
votes
0answers
172 views

A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that: For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index ...
2
votes
1answer
126 views

k-fellow traveler property and automatic structur

Let P be a permutation group with some generating set S and let W be the word acceptor automaton of P, if I know the value of k (k-fellow-traveller property of CayleyGraph CG(P,S)). I realized that ...