# Questions tagged [geometric-group-theory]

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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### Examples of "Monster" groups

I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are: 1.) Non-...
67 views

### Estimate word-metric length in free nilpotent groups

I would like to estimate the length of a word in a free nilpotent group. As the first example, I would like to estimate the word metric in the Heisenberg group $H_3$. This is the group of upper ...
281 views

### On the homological dimension of a Borel construction

Let $M$ b a closed connected smooth manifold with fundamental group $\Gamma$. Suppose $G$ is a simply-connected Lie group that acts smoothly on $M$. Then the Borel construction $$M//G = M \times_G EG$$...
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### Coarse quotient maps

Interesting connections and analogies have been observed between non-linear geometry of Banach spaces and coarse geometry. In the former subject, people have investigated the notion of uniform (or ...
108 views

### Uniform amenability at infinity

Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$ there is a finite subset $F\subset G$ such that \max_{s\in E} |s F \mathbin{\triangle} F| \le \...
621 views

### Examples of results first proved using geometrical methods?

Hi all, I am beginning to learn about geometric group theory. I would like to write a little exposé intended to be read by the uninitated, so it would be nice to talk about (preferably simple) ...
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### Your favorite papers on geometric group theory

I would like to improve my "depth of understanding" in geometric group theory. So I am interested in short and accessible papers on subjects related to this field but which are not always ...
155 views

### Quasi-isometric rigidity of surface groups and commensurability

Let $G$ be a group quasi-isometric to the fundamental group of a genus 2 surface group $H$. It is well known that $G$ is quasi-isometrically rigid, i.e. $G$ and $H$ are virtually isomorphic. Does ...
325 views

### Decidability of word problem for group admitting certain action

Let $G$ be a group acting highly transitively (and faithfully) on a set $S$. Suppose that $G$ is finitely presented, and that every stabilizer in $G$ of a finite subset of $S$ is finitely generated. I ...
299 views

### Contractible Rips complex from non-hyperbolic group

I heard that the Rips complexes associated to the Cayley graphs of hyperbolic groups are contractible for a sufficiently large radius. Is the converse true? Namely, if a group is non-hyperbolic, then ...
522 views

### Amenable action intuition

Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
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### Non-compact Dirichlet fundamental domains and free Fuchsian groups

Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model. Assume throughout that $\mathcal{F}$...
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### Let $G$ be an accessible f.g. group, then can we bound the number of vertex orbits in a reduced $G$-tree with finite edge-stabilisers?

I'm currently working on understanding the accessibility results of Dunwoody  and Bestvina-Feighn . Both of these papers seem to state different Dunwoody's accessibility result differently. In ...
2k views

### Equivalent definitions of Gromov hyperbolicity

Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible. I'm happy for the definitions to require some niceness ...
89 views

### What are the order 5 symmetries of the the torus knot T(3,4)?

Generally, I am interested in understanding the free actions of finite cyclic groups on $S^3$ which leave invariant an oriented torus knot $T(p,q)$. For a specific example, consider the knot $T(3,4)$ ...
69 views

### The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic

Given a group $G$, suppose $G$ admits a non-elementary acylindrical action on a Gromov hyperbolic space $S$. I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically ...
81 views

### Is $\mathbb Z$ a subgroup of a nontopologizable polybounded countable group?

A group $G$ is called $\bullet$ topologizable if $G$ is algebraically isomorphic to a non-discrete Hausdorff topological group; $\bullet$ nontopologizable if $G$ is not topologizable; $\bullet$ ...
136 views

### Do compact universal covers have concentration of measure phenomenon?

$\DeclareMathOperator\vol{vol}\DeclareMathOperator\diam{diam}$I have a sequence of compact Riemannian manifolds $M_n$ with $\diam (M_n) \to 0$ and finite fundamental groups $\pi_1 (M_n)$ so that their ...
276 views

### Subgroups of RAAGs vs. subgroups of RACGs

Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group? It is well-known from the theory of special cube complexes that ...
80 views

### Examples of nonlinear residually finite hyperbolic groups

What are some examples of nonlinear residually finite hyperbolic groups?
498 views

### Fundamental group of $M_g^\circ$

Let's work over the complex numbers $\mathbb{C}$. Let $g\geq3$ be an integer. Let $\mathcal{M}_g$ be moduli stack of smooth genus $g$ curves. Let $M_g$ be the corresponding coarse moduli scheme. They ...
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### Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?

As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms. There are a relationship between the mapping class ...
274 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 (...
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### Mapping class groups of Haken Seifert 3-manifolds (not small)

This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it. I want to ...
247 views

### Characterizations of metric trees

Let $X$ be a geodesic space. Then the following conditions are equivalent: For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$. No ...
137 views

### Realizing groups as the fundamental group of graphs of groups allowing non-injective maps?

In the definition of a graph of groups it is assumed that the maps from the edge groups to the vertex groups of injections, however in what follows I will also be interested in the case where the maps ...
316 views

### Virtually large groups of small rank (related to 3-manifolds)

Edited 25.05.21: the assumptions of the question were incorrect, but as the discussion may be helpful for future MOnauts, I'll strike my mistakes and add clearly marked explanations afterwards. I am ...
111 views

### Amenable automatic groups

Are there any known examples of finitely generated groups that are both amenable and automatic, besides the easy example of virtually abelian groups? Or are there any known restrictions that arise if ...
210 views

### Some questions on a paper of Baumslag and Solitar

I was recently trying taking a look at the paper "Some two-generator one-relator non-hopfian groups" by Baumslag and Solitar where they introduce the groups now known as the Baumslag-Solitar ...
193 views

### Geometric intuition behind Garside's paper?

I apologize in advance for a somewhat wishy-washy question. I just read the paper "The Braid Group and Other Groups" by F. A. Garside in which he solves the conjugacy problem for the braid ...
486 views

### Cantor-Bernstein for quasi-isometric embeddings?

Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric ...
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### Ping pong with parabolic isometries on Gromov hyperbolic spaces

For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
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### Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether ...
215 views

### Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?

Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs). Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...
118 views

### Infinite oscillation of minimum word length in 2-generated group

Let $G$ be a group with generators $a, b\in G$. Define $\mathrm{len}:G\to\mathbb{Z}_{\geq 0}$ by sending $g$ to the minimum length of a word in $a, b, a^{-1}, b^{-1}$ equal to $g$. Assume that for all ...
49 views

### Quasi-isometry of solvable minimax groups

[Edits in brackets] Consider two finitely generated solvable minimax groups $G_i$ ($i = 1,2$) so that $1 \to N_i \to G_i \to Z_i \to 1$ [splits] with $N_i$ nilpotent, $Z_i$ infinite cyclic and $G_i$ ...
367 views

### Membership to double cosets in free groups

Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group? Has this membership problem been implemented in GAP/Magma? More ...
142 views

### Is it possible for a finitely generated group to have polynomial growth rate with leading coefficient less than 1?

I'm quite a neophyte in this area, so apologies if this question is easy. (I've edited slightly to make my question more clear.) I've been trying to learn about growth rates for finitely generated ...
212 views

### Groups acting on products of hyperbolic spaces

I am interested in groups acting properly and cocompactly by isometries on finite products of Gromov-hyperbolic metric spaces. I am mostly interested in the case where the group itself is not ...
60 views

### When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open. Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace. Under what ...
207 views

### Is the automorphism group of free group of rank two relatively hyperbolic?

By Behrstock, Drutu and Mosher [BDM], we know that the (outer) automorphism groups $\mathrm{Aut}(F_n)$ and $\mathrm{Out}(F_n)$ of free group of rank $n$ are not relatively hyperbolic if $n \geq 3$ (...
103 views

### Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure. Marc Peigné  constructed examples of geometrically infinite hyperbolic ...
125 views

### Characterization of Freudenthal (end) compactification

I had seen somewhere in the literature that the Freudenthal compactification of a locally compact, connected, locally connected, $\sigma$-compact, Hausdorff topological space $X$, is the maximal ideal ...
There is a procedure, suggested by Vaughan Jones, which associates a link to every element of Thompson's group F. Also every knot or link in $\mathbb{R}^3$ can be obtained in this way. A subclass of ...