**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 ...

**26**

votes

**4**answers

1k 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 ...

**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 ...

**22**

votes

**9**answers

4k views

### Introductory text on geometric group theory?

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

**22**

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?
...

**20**

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 ...

**20**

votes

**2**answers

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 ...

**18**

votes

**3**answers

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 ...

**16**

votes

**4**answers

718 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 ...

**16**

votes

**1**answer

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 ...

**16**

votes

**1**answer

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 ...

**15**

votes

**4**answers

740 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 ...

**15**

votes

**3**answers

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 $\pi_2(X)\ne 1$?

**14**

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?

**14**

votes

**1**answer

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 ...

**13**

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 ...

**13**

votes

**2**answers

446 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

435 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 ...

**12**

votes

**3**answers

746 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

345 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

164 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

**0**answers

271 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 ...

**11**

votes

**3**answers

657 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

**1**answer

272 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 ...

**11**

votes

**2**answers

458 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 ...

**11**

votes

**2**answers

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 ...

**11**

votes

**1**answer

772 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

**0**answers

291 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 ...

**10**

votes

**6**answers

849 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 ...

**10**

votes

**4**answers

849 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

**2**answers

204 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 ...

**10**

votes

**1**answer

227 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

197 views

### Growth of Poincare duality groups

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

**10**

votes

**2**answers

336 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

**2**answers

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 ...

**10**

votes

**3**answers

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.) ...

**10**

votes

**1**answer

424 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 ...

**10**

votes

**0**answers

485 views

### Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...

**9**

votes

**3**answers

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 ...

**9**

votes

**1**answer

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...

**9**

votes

**1**answer

293 views

### Decidability of conjugacy problem for finitely generated subgroups of free groups

The conjugacy problem for a free group $F_n$ on $n$ letters has an easy solution. Each element of $F_n$ is conjugate to a unique and easily computable "cyclically reduced element" (this means that if ...

**9**

votes

**0**answers

178 views

### Group with unsolvable conjugacy problem but solvable conjugacy length?

Could there exist a finitely presented group with unsolvable conjugacy problem, in which it is decidable whether a word over the group generators is a shortest representative of an element in its ...

**8**

votes

**2**answers

331 views

### Converse to Milnor's theorem on manifolds with nonnegative Ricci curvature.

Disclaimer : I suspect the question I am about to ask is really hard, but I just want to know the status of such questions.
Thanks to Milnor, we know that the $\pi_1$ any compact manifold with ...

**8**

votes

**2**answers

387 views

### Which groups have nice compactifications ?

Given a discrete group G. Is there a nice criterion to decide, whether there is a compact Hausdorff $G$- space X, that contains the discrete space $G$ as a subspace, such that the stabilizer of every ...

**8**

votes

**1**answer

312 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 ...

**8**

votes

**1**answer

140 views

### Counterexamples to analogue of Cannon conjecture in higher dimensions

It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for ...

**8**

votes

**1**answer

196 views

### Topology of boundaries of hyperbolic groups

For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...

**8**

votes

**1**answer

236 views

### Are Hyperbolic Groups Residually Amenable

It is a well-known conjecture (or maybe just a question) that all hyperbolic groups are residually finite. What happens if we weaken the conclusion; in particular
Are all hyperbolic groups residually ...

**8**

votes

**2**answers

633 views

### Actions of Thompson group F

Does anybody know the actions of Thompson group F which are not conjugate to the standard one?
Motivation is to find actions such that the Schreier graph of the action does not contain a binary tree.
...

**8**

votes

**1**answer

159 views

### Do finitely generated groups of polynomial growth satisfy a “uniform covering property?”

Let $G$ be a finitely generated discrete group with a finite symmetric generating set $S=S^{-1}\subset G$. For every group element $g$, define $\|g\|_S$ to be the length with respect to $S$, i.e. the ...