Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations
0
votes
0answers
31 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
2answers
194 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 ...
2
votes
0answers
58 views
Does there exists a finitely presented group with Dehn function > n^3 and all asymptotic cones simply connected
It is well known that all asymptotic cones simply connected implies polynomial Dehn function (Gromov). It is also well known that quadratic Dehn function implies all asymptotic cones simply connected ...
0
votes
0answers
132 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 ...
3
votes
1answer
94 views
Non co-hopfian groups - finite index
What sort of spaces cover themselves with a finite fibre? Or, what sort of finitely generated groups contain isomorphic copies of themselves as subgroups as finite index? Is it a reasonable question ...
6
votes
1answer
133 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 ...
2
votes
0answers
206 views
Fundamental group of a Cayley graph [migrated]
Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...
0
votes
0answers
82 views
Help understanding a proof in Stallings' Triangle of Groups paper
I'm trying to understand the proof of theorem 1 in Stalling's Non-positively Curved Triangle of Groups. I have specific questions, but is there anywhere someone has written out the proof in more ...
1
vote
0answers
62 views
Is there any study on limit sets arising from “outside” of the hyperbolic geometry?
In the projective model, hyperbolic spaces are modeled by the interior of the light cone of Lorentz spaces. Kleinian groups are isometries of the hyperbolic space, and they centainly also act on the ...
4
votes
1answer
96 views
Connection between degree of growth and return probabilities of random walks on Lie groups
Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...
6
votes
0answers
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
0answers
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
1answer
142 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
1answer
201 views
Candidates for non-sofic groups
What are the "simplest" examples of countable groups that are not known to be sofic?
3
votes
0answers
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
1answer
174 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
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 ...
1
vote
1answer
86 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
166 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
1answer
124 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 = ...
2
votes
2answers
185 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
135 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
122 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
67 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 ...
7
votes
1answer
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
0answers
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
0answers
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 ...
4
votes
1answer
92 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 ...
2
votes
1answer
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
0answers
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 ...
5
votes
1answer
146 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
227 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 ...
3
votes
3answers
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
2answers
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 ...
1
vote
1answer
120 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...) ?
12
votes
0answers
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
1answer
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
1answer
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
1answer
293 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
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 ...
5
votes
1answer
123 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
250 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
120 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
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 ...
0
votes
1answer
162 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
140 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
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?
2
votes
1answer
157 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
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
1answer
270 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 ...
