**3**

votes

**1**answer

328 views

### Automorphisms of Hyperbolic groups and Graphs of Groups

I have been reading Levitt's paper Automorphisms of Hyperbolic groups and Graphs of Groups. I am having some trouble trying to fit all the bits together, and would appreciate some help with this last ...

**5**

votes

**0**answers

144 views

### Dynamics of virtual automorphisms of free group

The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism.
Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the ...

**7**

votes

**0**answers

410 views

### Uniquely geodesic groups

Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...

**0**

votes

**1**answer

188 views

### A question on Cayley graphs and hyperbolic 3-manifolds

There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...

**8**

votes

**1**answer

160 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 ...

**5**

votes

**1**answer

281 views

### Fixed points on boundary of hyperbolic group

Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?

**2**

votes

**1**answer

191 views

### Group actions on trees and translates under hyperbolic elements

I have the following question regarding group actions on trees to which I suspect the answer to be "yes", but it could very well be that extra conditions are required (it is certainly true for free ...

**7**

votes

**2**answers

605 views

### Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?

Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , ...

**12**

votes

**1**answer

312 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 ...

**6**

votes

**1**answer

388 views

### Partition of a group into small subsets

A nonempty subset $S$ of a group $G$ is called small if there is an infinite sequence of elements $g_n$ in $G$ such that the translated sets $g_nS$ are pairwise disjoint.
Question: Is there a group ...

**11**

votes

**1**answer

574 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 ...

**12**

votes

**0**answers

307 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 ...

**9**

votes

**0**answers

220 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 ...

**12**

votes

**2**answers

847 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 ...

**2**

votes

**0**answers

112 views

### When does order matter when decomposing a boundedly generated group

A group $G$ is said to be boundedly generated if (it is finitely generated and) there exists a finite family of cyclic subgroups (not necessarily normal or distinct) $\lbrace C_i \rbrace_{i =1, ...

**8**

votes

**1**answer

277 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 ...

**10**

votes

**1**answer

221 views

### Growth of Poincare duality groups

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

**6**

votes

**1**answer

191 views

### Lyndon-Schützenberger for torsion-free hyperbolic groups

Given a torsion-free hyperbolic group $G$, does there exist a number $n(G)$ such that for any $x,y,z\in G$, $x^n y^n z^n =1$ implies that $x$, $y$, and $z$ commute pairwise?
Some musings/questions...
...

**7**

votes

**1**answer

408 views

### Quantum Cellular Automata on Riemannian manifolds and geometric group theory

We try to motivate our question. We have a certain logical/operational structure that has an
emergent physical interpretation. We are giving this structure a geometric setting via
quasi-isometries. ...

**12**

votes

**1**answer

473 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 ...

**0**

votes

**0**answers

303 views

### Proper Group action on a metric space

Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ ...

**12**

votes

**3**answers

786 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 ...

**8**

votes

**1**answer

381 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 ...

**1**

vote

**1**answer

117 views

### Amenable normal closure

Prove or disprove:
Let $G$ be a countable group. Let $H < G$ be an amenable subgroup with a finite conjugacy class. Then the normal closure of $H$ is also amenable.
Thanks!

**3**

votes

**3**answers

477 views

### Group action on the real line

Hi,
I was wondering about the following question:
if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct a new action such ...

**6**

votes

**1**answer

459 views

### Growth of Thompson's group $F$

EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with ...

**4**

votes

**1**answer

231 views

### Cayley graphs of finitely generated infinite groups quasi-isometrically embeddable in R^3

Dear friends,
I am only a theoretical physicist. However, the answer to this question is relevant for emergence of space-time from a quantum cellular automaton (in the future I will pose a much more ...

**11**

votes

**3**answers

860 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 ...

**3**

votes

**1**answer

340 views

### Actions of Thompson group F. II

Let $\Gamma$ be a group generated by symmetric finite set $S$ and acting on $X$. The Schreier graph of the action is the graph with vertex set $X$ and $(x,y)$ is an edge if there is $s\in S$ such that ...

**12**

votes

**0**answers

328 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 ...

**8**

votes

**2**answers

362 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 ...

**4**

votes

**1**answer

178 views

### Periodic automorphisms of free groups and surface homeomorphisms

Denote by $F_n$ the free group of rank $n$. We say that an automorphism $\phi\in Aut(F_n)$ is geometric if there exists a surface with boundary $M$ and a homeomorphism $h\colon M\to M$ such that $h$ ...

**8**

votes

**2**answers

691 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.
...

**7**

votes

**1**answer

241 views

### Unbounded metrics on groups

If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?

**2**

votes

**1**answer

132 views

### Polynomial growth of the Betti number of balls of the Cayley graphs

Consider a finitely generated group. Assume that the first Betti number of the ball of radius n in the Cayley graph is at most polynomial in n. This property is satisfied by free groups and groups of ...

**5**

votes

**0**answers

165 views

### Can one characterize amenable groups with $C_G(x)$ cyclic for all $x\neq 1$?

Can one characterize the amenable groups that have the property that the centralizer of every non-identity element is cyclic?
For example, must they be solvable?

**4**

votes

**2**answers

342 views

### Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators?

We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set.
I have been constructing a space ...

**5**

votes

**1**answer

331 views

### Thompson's group T

Does there exist a non trivial homomorphism from Thompson's group T to a linear group?

**0**

votes

**1**answer

240 views

### Ordered groups - examples

Let $G=BS(m,n)$ denote the Baumslag–Solitar groups defined by
the presentation $\langle a,b: b^m a=a b^n\rangle$.
We assume that G is non-abelian.
Question: Find $m,n$ such that $G$ is an ...

**13**

votes

**2**answers

600 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 ...

**3**

votes

**0**answers

100 views

### Cogrowth and value of its series at the critical exponent

Let $G$ be a finitely generated group and write $G = F/N$ for $N$ a normal subgroup of a free group $F$. Let $S_n$ be the elements in $F$ written as words of exactly $n$ letters. So, for $n\geq 1$, ...

**7**

votes

**2**answers

484 views

### Are virtual cubulated groups cubulated?

Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex?
Edit: After ...

**1**

vote

**1**answer

554 views

### Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...

**6**

votes

**1**answer

405 views

### Classification of geometric outer automorphisms of free groups

Good evening everyone,
an outer automorphism $[\phi]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism $h\colon M\stackrel{\cong}{\to}M$, where $M$ is a compact surface with ...

**10**

votes

**0**answers

532 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 ...

**7**

votes

**1**answer

254 views

### How large is this “algebra” of defining graphs for Right-angled Artin groups?

As part of my research, I have been trying to construct a spherical space at infinity for every right-angled artin group. I've been able to work it out for a certain class of defining graphs. I'd like ...

**3**

votes

**1**answer

289 views

### On Canonical generators of torsion free nilpotent group

I read somewhere that Maltcev proved that for any finitely generated torsion free Nilpotent group $G$ there are canonical generators, i.e.
$g_{1},\ldots,g_{k}$ such that any $g \in G$ can be written ...

**8**

votes

**0**answers

394 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 ...

**8**

votes

**4**answers

482 views

### isometric embeddings of Cayley graphs in “nice” spaces

This is from a physicist I know and as may be expected, I am threading my way between poorly defined and poorly translated.
What groups have Cayley graphs (w.r.t. a fixed finite generating set, and ...

**7**

votes

**1**answer

194 views

### Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups

I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions.
$G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ ...