# Tagged Questions

**3**

votes

**0**answers

61 views

### Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...

**10**

votes

**2**answers

205 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 ...

**0**

votes

**0**answers

133 views

### Induced graphs of cayley graph

I have a Cayley graph $Cay(G,S)$, its group presentation $G=< S | R >$ and it is a metric graph by assigning a length equal to 1 to each edge. I also have an induced subgraph of that Cayley ...

**6**

votes

**0**answers

96 views

### How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...

**0**

votes

**0**answers

69 views

### Groups and triangle-square complexes

I would like to know what kind of groups (and/or their group presentation) acting geometrically on CAT(0) curved piecewise Euclidean triangle-square complexes.
Thanks

**1**

vote

**1**answer

143 views

### JSJ-decompositions of hyperbolic groups and elementary vertices

My question is the following:
In Bowditch's JSJ-decomposition of hyperbolic groups, can elementary (virtually-cyclic) vertices have degree 1? If not, why not?
I had thought for a long time that ...

**5**

votes

**1**answer

202 views

### Candidates for non-sofic groups

What are the "simplest" examples of countable groups that are not known to be sofic?

**3**

votes

**0**answers

101 views

### Computing the pro-solvable closure of a finitely generated subgroup of a free group

The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...

**8**

votes

**1**answer

175 views

### Is there a finitely generated residually finite group with solvable word problem that does not embed in a finitely presented residually finite group?

The famous Higman embedding theorem says that every recursively presented group embeds in a finitely presented group. This is a convenient tool to construct finitely presented groups with bizarre ...

**3**

votes

**0**answers

115 views

### Topological interpretation for groups of type $FP_2$

A group $G$ is of type $FP_2$ if it admits a partial projective resolution of $\mathbb{Z}G$-modules $$ P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0$$ with each $P_i$ being ...

**7**

votes

**1**answer

167 views

### Minimal normally generating subsets of minimal generating sets

Let $G$ be a finitely generated group. The weight $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called ...

**2**

votes

**1**answer

125 views

### Commutator width of soluble Baumslag Solitar groups

Do the soluble Baumslag-Solitar groups have finite commutator width? A soluble Baumslag-Solitar group is given by a presentation of the from
BS(1,m) = $<a,b \mbox{ }| \mbox{ } a^{-1}ba = ...

**5**

votes

**1**answer

136 views

### Commutator Width of a direct limit of hyperbolic groups

Is it known if the direct limit of hyperbolic groups can have finite commutator width? Every hyperbolic group has infinite verbal width for any word $w$, so in particular for the commutator word ...

**7**

votes

**1**answer

305 views

### How to construct a group with specified growth function

Are there any procedures which given a nonnegative nondecreasing function on the integers will construct a finitely generated group with the same growth up to the usual equivalence of growth ...

**2**

votes

**0**answers

148 views

### quasiisomorphic groups and torsion [closed]

Are there two finitely generated quasiisomorphic groups $G$ and $H$ such that $G$ is torsionfree and $H$ has torsion elements of arbitrarily large order?

**2**

votes

**0**answers

70 views

### Characterizations of product groups under quasi-isometry

This is a follow-up question to Quasi-isometric rigidity of certain products of groups. It can be moved to a comment if it is too trivial.
If all the asymptotic cones of a finitely generated group ...

**2**

votes

**1**answer

112 views

### Density of ends of long words in a hyperbolic group

Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...

**5**

votes

**0**answers

118 views

### Uniform incentre of collection of quasi convex subspaces in hyperbolic spaces

I'm reading a paper of Wise on cubulations and the following fact is used:
Let $H$ be a quasi-convex subgroup of a $\delta$-hyperbolic group and let $H_i, i\in I$ be a finite family of translates of ...

**3**

votes

**3**answers

217 views

### Domination of length functions of trees with equal covolume

(This is a reformulation of an earlier unanswered question. I would like to thank Ian Agol for pointing out to me Walter Parry's characterization of hyperbolic translation length functions.)
Let $G$ ...

**6**

votes

**2**answers

374 views

### Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...

**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 ...

**5**

votes

**1**answer

137 views

### distortion of cyclic subgroups of linear groups

In an informal talk I heard a statement:
"Any cyclic subgroup in a linear group is at most exponentially distorted"
with a vague reference to a work of Lubotzky with coauthors.
The works of ...

**4**

votes

**1**answer

188 views

### Interesting families of groups as group extensions

Let me start this question with an example that hopefully makes clear what I am looking for:
A discrete subgroup $G$ of the group of euclidean isometries of $\mathbb{R}^d$ is called a ...

**7**

votes

**1**answer

294 views

### Analogues of the curve complex for Out(F)

Let $F$ be a finitely generated free group.
Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be ...

**8**

votes

**0**answers

156 views

### A Magnus theorem in the category of residually finite groups

There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle ...

**3**

votes

**1**answer

251 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

121 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

366 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 ...

**5**

votes

**1**answer

262 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?

**7**

votes

**2**answers

599 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 , ...

**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 ...

**6**

votes

**1**answer

364 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 ...

**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 ...

**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 ...

**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 ...

**2**

votes

**0**answers

110 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, ...

**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

**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 ...

**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 ...

**6**

votes

**1**answer

421 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 ...

**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 ...

**3**

votes

**1**answer

306 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 ...

**11**

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**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 ...

**4**

votes

**1**answer

164 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

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.
...

**7**

votes

**1**answer

232 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

119 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

158 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

237 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

306 views

### Thompson's group T

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