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

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3
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1answer
114 views

Quasi-isometric rigidity of certain products of groups

Let $G_1,G_2$ be two delta-hyperbolic groups. If $G_1\times \Bbb{Z}$ is quasi-isometric to $G_2\times \Bbb{Z}$, must it be true that $G_1$ is quasi-isometric to $G_2$? This is similar to the classic ...
1
vote
1answer
141 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 ...
4
votes
0answers
126 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 ...
5
votes
1answer
178 views

Hilbert space compression of the lamplighter group

What is the Hilbert space compression exponent of the standard lamplighter group $\mathbb{Z_{2}} \wr \mathbb{Z}$? For $\mathbb{Z} \wr \mathbb{Z}$ it is known to be $2/3$ by work of Austin, Naor and ...
10
votes
1answer
338 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 ...
4
votes
3answers
244 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 $G$ ...
7
votes
2answers
717 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 ...
1
vote
1answer
165 views

Quasi-isometric embeddings of the mapping class group into the Teichmuller space

Does there exist a quasi-isometric embedding $$MCG(S) \to (\mathrm{Teich}(S), d)$$ for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?
13
votes
0answers
218 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
1answer
215 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 ...
5
votes
1answer
211 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
1answer
351 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
0answers
192 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 ...
5
votes
1answer
143 views

Injective simplicial maps between Arc complexes

Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...
3
votes
1answer
329 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
0answers
146 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
0answers
417 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
1answer
190 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
1answer
162 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
1answer
284 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
1answer
192 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
2answers
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
1answer
315 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
1answer
391 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
1answer
601 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
0answers
310 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
0answers
225 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
2answers
880 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
0answers
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
1answer
287 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
1answer
222 views

Growth of Poincare duality groups

Can one prove that Poincare duality groups cannot have intermediate growth?
6
votes
1answer
196 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
1answer
412 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
1answer
488 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
0answers
311 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
3answers
797 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
1answer
397 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
1answer
118 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
3answers
482 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
1answer
468 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
1answer
234 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
3answers
907 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
1answer
343 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
0answers
331 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
2answers
365 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
1answer
181 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
2answers
696 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
1answer
243 views

Unbounded metrics on groups

If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?
2
votes
1answer
135 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
0answers
166 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?