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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7
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1answer
629 views

Candidates for non-sofic groups

What are the "simplest" examples of countable groups that are not known to be sofic?
5
votes
0answers
321 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 ...
9
votes
1answer
295 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
0answers
142 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 ...
1
vote
1answer
133 views

Module structure of the abelianization of the commutator subgroup

Let $G$ be a (non-abelian) group, and let $G_2$ denote its commutator subgroup. Then the abelianization $G_2^{ab} = H_1(G_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The ...
7
votes
1answer
214 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
2answers
257 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 = ...
3
votes
2answers
312 views

Action of Mapping Class Group on Arc complex

Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...
5
votes
1answer
175 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 ...
2
votes
1answer
151 views

Can a tree cover a finite graph with arbitrary large girth?

Let $G$ be some discrete finitely generated group acting cocompactly on a leafless tree $T$. Is it true that for any natural number $n$ there is a finite graph $\Gamma$ such that: $T$ is the universal ...
0
votes
0answers
84 views

What is the definition of an HNN extention of a group, relative to another group?

I am familiar with the definition of the HNN extension of a group relative to an isomorphism between two of its subgroups. For comparison's sake let me make that explicit. For groups $G_1, G_2\leq ...
9
votes
1answer
382 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
0answers
152 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
0answers
88 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 ...
3
votes
1answer
131 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
148 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
127 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 ...
9
votes
2answers
568 views

Folner sets and balls

Several related questions were asked before on MO, but it is not clear to me if the following was settled. Given a finitely generated amenable group, is it always possible to find some finite ...
5
votes
1answer
185 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 ...
13
votes
1answer
443 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
252 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 $G$ ...
10
votes
2answers
835 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
180 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
238 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
269 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
221 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 ...
8
votes
1answer
381 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
204 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
149 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
377 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 ...
6
votes
0answers
154 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 ...
8
votes
0answers
446 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
205 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
168 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
296 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
206 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
606 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
326 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
419 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 ...
12
votes
1answer
749 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
322 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
265 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 ...
13
votes
2answers
1k 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
118 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
371 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 ...
11
votes
1answer
267 views

Growth of Poincaré duality groups

Can one prove that Poincaré duality groups cannot have intermediate growth?
6
votes
1answer
212 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
447 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. ...
13
votes
1answer
590 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
405 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 $U=$ ...