**31**

votes

**1**answer

2k views

### orders of products of permutations

Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...

**30**

votes

**4**answers

2k views

### Is there a simple proof that a group of linear growth is quasi-isometric to Z?

I proposed to a master's student to work, from the exercise in Ghys-de la Harpe's book, on the proof that a finitely generated group $G$ that is quasi-isometric to $\mathbb{Z}$ is virtually ...

**25**

votes

**9**answers

5k views

### Introductory text on geometric group theory?

Can someone indicate me a good introductory text on geometric group theory?

**24**

votes

**4**answers

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?
...

**24**

votes

**2**answers

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 ...

**20**

votes

**2**answers

859 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 ...

**19**

votes

**3**answers

1k views

### An example of a non-amenable exact group without free subgroups.

A countable discrete group $\Gamma$ is said to be exact if it admits an amenable action on some compact space.
So clearly amenable groups are exact, but large familes of non-amenable groups are as ...

**18**

votes

**6**answers

1k views

### Residual finiteness: why do we care?

Residually finite groups have been studied for a long time. However, I am struggling to work out why we care, or perhaps, why they continue to be of interest. Let me explain.
Magnus, in his 1968 ...

**18**

votes

**3**answers

811 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 ...

**17**

votes

**4**answers

813 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 ...

**17**

votes

**1**answer

452 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 ...

**16**

votes

**4**answers

821 views

### Free splittings of one-relator groups

Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings.
Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is ...

**16**

votes

**1**answer

728 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 ...

**15**

votes

**9**answers

3k views

### The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$)

Is there someone who can give me some hints/references to the proof of this fact?

**15**

votes

**3**answers

1k 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 $\pi_2(X)\ne 1$?

**15**

votes

**1**answer

345 views

### Asymmetric metrics and cohomology

If $(X,d)$ is a metric space and $f : X \rightarrow \mathbb{R}$ is a Lipschitz function with Lipschitz constant $k < 1$, then the function
$$
D(x,y) := d(x,y) + f(y) - f(x)
$$
defines an asymmetric ...

**14**

votes

**4**answers

1k views

### Braid groups acting on CAT(0)-complexes

Does the braid group $B_n, n\ge 3$, act properly by isometries on a CAT(0) cube complex?
Update 1. During a recent talk of Nigel Higson in Pennstate Dmitri Burago asked whether the braid groups are ...

**14**

votes

**3**answers

503 views

### Failure of Mostow rigidity in dim. 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 $\mathbb H^2$ ...

**14**

votes

**1**answer

562 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 ...

**13**

votes

**2**answers

579 views

### Normal subgroups of finite index in free groups

Hi all,
This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of all normal subgroups ...

**13**

votes

**2**answers

346 views

### Minimal number of generators for $GL(n,\mathbb{Z})$

$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = ...

**13**

votes

**2**answers

471 views

### The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...

**13**

votes

**0**answers

269 views

### Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...

**13**

votes

**0**answers

210 views

### Bounding the lengths of the conjugators in the word problem for finite group presentations

Let $G = \langle X \mid R \rangle$ be a group defined by a finite presentation, and let $F$ be the free group on $X$. If $w \in F$ represents the identity in $G$, then $w$ is equal in $F$ to (the free ...

**12**

votes

**1**answer

307 views

### Is the Thompson group F locally indicable?

A group $G$ is called locally indicable if for any finitely generated subgroup $H \subset G$, there is a non-trivial homomorphism from $H$ to the real additive group $(\mathbb{R},+)$.
Is the Thompson ...

**12**

votes

**2**answers

760 views

### The fundamental group of a closed surface without classification of surfaces?

The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...

**12**

votes

**3**answers

781 views

### How can I tell if a group is linear?

The basic question is in the title, but I am interested in both necessary and sufficient conditions.
I know the Tits' alternative and Malcev's result that finitely generated linear groups are ...

**12**

votes

**1**answer

454 views

### Distortion of malnormal subgroup of hyperbolic groups

Let $G$ be a countable, Gromov-hyperbolic group.
We say that $H$ is hyperbolically embedded in $G$ if $G$ is relatively hyperbolic to {$H$} (in the strong sense). This definition is due to Osin.
A ...

**12**

votes

**0**answers

301 views

### What is an example of a word hyperbolic group without a finite complete rewriting system?

I believe that it was an open question back when I was a graduate student whether every word hyperbolic group admits a finite complete (=Church-Rosser=Noetherian+confluent) rewriting system for some ...

**12**

votes

**0**answers

322 views

### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...

**11**

votes

**6**answers

1k 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 ...

**11**

votes

**3**answers

816 views

### The role of the Automatic Groups in the history of Geometric Group Theory

What is the role of the theory of Automatic Groups in the history of Geometric Group Theory?
Motivation:
When I read through the "Word Processing in Groups" I was amazed by the supreme beauty and ...

**11**

votes

**2**answers

329 views

### Algorithms in hyperbolic groups

I'm stuck in some algorithms in hyperbolic groups, which may be rather simple.
Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can ...

**11**

votes

**2**answers

499 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 ...

**11**

votes

**1**answer

295 views

### What are smallest finite images of triangle groups?

Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...

**11**

votes

**3**answers

2k 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.) ...

**11**

votes

**1**answer

829 views

### Topological HNN extensions

First, let me recall what an abstract HNN extension is. Let $G$ be an abstract group, $A, B < G$ be subgroups of $G$ and $\phi : A \to B$ be an isomorphisms. Then there is a group $H$ and an ...

**11**

votes

**1**answer

348 views

### Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?

**10**

votes

**4**answers

995 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 ...

**10**

votes

**1**answer

309 views

### Mapping class group and CAT(0) spaces

I hope the questions are not too vague.
Is the mapping class group of an orientable punctured surface $CAT(0)$ ?
Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...

**10**

votes

**1**answer

218 views

### Growth of Poincare duality groups

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

**10**

votes

**2**answers

375 views

### Invariant free factor of a free group

Let $F_n=F\ast F'$ be a free splitting of the free group $F_n$ and $\phi\in Aut(F_n)$. The free factor $F$ is said to be invariant under $\phi$ if $\phi(F)\subseteq F$.
I recently wondered if this ...

**10**

votes

**1**answer

540 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 ...

**10**

votes

**2**answers

471 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 ...

**10**

votes

**1**answer

188 views

### Cantor-Bernstein for quasi-isometric embeddings?

Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric ...

**10**

votes

**1**answer

377 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...

**10**

votes

**1**answer

195 views

### Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...

**10**

votes

**1**answer

375 views

### Holomorphic cusp forms and cohomology of GL(2,Z)

Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...

**10**

votes

**1**answer

237 views

### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...

**10**

votes

**1**answer

445 views

### Partial word orders on groups

This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have ...