# Questions tagged [geometric-group-theory]

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

735
questions

**42**

votes

**13**answers

10k views

### Introductory text on geometric group theory?

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

**40**

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 ...

**36**

votes

**4**answers

3k 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 $\mathbb{Z}...

**32**

votes

**3**answers

2k views

### Is the Hurewicz theorem ever used to compute abelianizations?

The Hurewicz theorem tells us that if $X$ is a path-connected space then $H_1(X, \, \mathbb{Z})$ is isomorphic to the abelianisation of $\pi_1(X)$. This gives a potential method for computing the ...

**32**

votes

**1**answer

1k views

### Is this conjecture strictly weaker than P=NP?

My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...

**29**

votes

**0**answers

781 views

### Is this representation of Go (game) irreducible?

This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...

**28**

votes

**4**answers

2k 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?
...

**28**

votes

**2**answers

2k 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 $[x,y,...

**27**

votes

**13**answers

1k views

### Equivalent definitions of Gromov hyperbolicity

Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible.
I'm happy for the definitions to require some niceness ...

**26**

votes

**1**answer

1k views

### Group with finite outer automorphism group and large center

Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is ...

**26**

votes

**1**answer

527 views

### What is the minimal dimension of a complex realising a group representation?

This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).
Many interesting integral ...

**25**

votes

**7**answers

3k 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 ...

**23**

votes

**1**answer

753 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 ...

**22**

votes

**5**answers

4k views

### Are there any computational problems in groups that are harder than P?

There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like ...

**22**

votes

**5**answers

1k 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 ...

**22**

votes

**2**answers

1k 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 ...

**21**

votes

**4**answers

1k 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 ...

**21**

votes

**4**answers

1k 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 it ...

**21**

votes

**2**answers

1k views

### Modern references on hyperbolic groups

Several good references dedicated to hyperbolic groups have been written until 1990, including:
Hyperbolic groups, written by M. Gromov.
Géométrie et théorie des groupes : les groupes hyperboliques ...

**21**

votes

**1**answer

586 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 ...

**21**

votes

**1**answer

635 views

### Can a hyperbolic, one ended, one relator group, have a shorter trivial word?

Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the ...

**20**

votes

**9**answers

7k 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?

**20**

votes

**3**answers

2k 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

**4**answers

1k views

### Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?
The best I could get by trial and error is an embedding ...

**20**

votes

**1**answer

873 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 ...

**19**

votes

**7**answers

3k 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 ...

**19**

votes

**2**answers

3k 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 = \begin{...

**19**

votes

**3**answers

1k 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 $\mathbb H^2$ ...

**19**

votes

**0**answers

736 views

### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...

**18**

votes

**3**answers

2k 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 ...

**18**

votes

**4**answers

2k views

### Units in the group ring over fours group after Gardam

Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$.
This conjecture asserted that if $\Gamma$ is a ...

**18**

votes

**1**answer

467 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 ...

**18**

votes

**0**answers

431 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 $F_2$. ...

**17**

votes

**7**answers

2k views

### Examples of residually-finite groups

One of the main reasons I only supervised one PhD student is that I find it hard to find an appropriate topic for a PhD project. A good approach, in my view, is to have on the one hand a list of ...

**17**

votes

**4**answers

2k 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 ...

**17**

votes

**1**answer

644 views

### Are there any “simple” monoids with intermediate growth?

The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...

**17**

votes

**1**answer

704 views

### Is Hopf property a quasi-isometry invariant?

Recall that a group $G$ is called Hopfian if every surjective endomorphism $G\to G$ is injective. Malcev observed that all finitely-generated (f.g.) residually finite groups are Hopfian. It is well-...

**17**

votes

**2**answers

724 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 ...

**17**

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 $\pi_2(X)\ne 1$?

**17**

votes

**0**answers

653 views

### How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...

**16**

votes

**3**answers

566 views

### Group with non-trivial center containing trivially-intersecting copies of itself

I'm trying to think of an example of a group $G$ with non-trivial center such that there exist subgroups $H_1,H_2\le G$ both isomorphic to $G$ and satisfying $H_1\cap H_2=\{1\}$. Does such a group ...

**16**

votes

**1**answer

366 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 $...

**16**

votes

**1**answer

302 views

### Existence of a quasi-isometric residually finite group?

It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...

**16**

votes

**2**answers

3k 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 ...

**16**

votes

**1**answer

808 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 ...

**16**

votes

**1**answer

440 views

### Is Thompson's group $T$ co-Hopfian?

A group $G$ is co-Hopfian if every injective homomorphism $G\to G$ is bijective, i.e., if $G$ contains no proper subgroups isomorphic to $G$. My question is whether Thompson's group $T$ is co-Hopfian.
...

**16**

votes

**0**answers

166 views

### Is there a simple group that is torsion-free, type $\textrm{F}_\infty$, and infinite dimensional?

Does there exist an example of a group that is:
Simple,
Torsion-free,
Of type $\textrm{F}_\infty$, and
Infinite dimensional (meaning of infinite cohomological dimension)?
Thompson's group $F$ has ...

**16**

votes

**0**answers

297 views

### Finitely generated groups with Hölder-exotic space of ends?

The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...

**15**

votes

**2**answers

724 views

### Is an HNN extension of a virtually torsion-free group virtually torsion-free?

This is a cross post from Math.StackExchange after 2 weeks without an answer and a bounty being placed on the question.
Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free ...

**15**

votes

**2**answers

2k 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 ...