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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12
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2answers
2k views

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-...
2
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0answers
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 ...
14
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1answer
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 $$M//G = M \times_G EG$$...
8
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0answers
100 views

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 ...
8
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0answers
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 \...
6
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4answers
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) ...
17
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3answers
1k views

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 ...
4
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1answer
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 ...
10
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2answers
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 ...
8
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2answers
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 ...
7
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2answers
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 ...
6
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1answer
173 views

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 $\mathcal{F}$...
4
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1answer
101 views

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 [1] and Bestvina-Feighn [2]. Both of these papers seem to state different Dunwoody's accessibility result differently. In ...
29
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16answers
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 ...
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0answers
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 $T(3,4)$ ...
3
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1answer
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 ...
3
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0answers
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; $\bullet$ ...
8
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0answers
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 ...
10
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3answers
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 ...
3
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0answers
80 views

Examples of nonlinear residually finite hyperbolic groups

What are some examples of nonlinear residually finite hyperbolic groups?
7
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2answers
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 ...
2
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0answers
70 views

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 ...
4
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2answers
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 (...
7
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1answer
265 views

Does any subset of a finitely generated group with positive upper density contain three points in arithmetic progression?

Let $G$ be a countable finitely generated group, with word metric $d_S$ induced by some generating set $S$. Let $B_r$ be the ball of radius $r$ around the identity under $d_S$. We say that $A \subset ...
14
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2answers
577 views

Prehistory of Gromov-hyperbolic spaces/groups

When speaking about hyperbolic groups/spaces, one usually refers to Gromov's monograph Hyperbolic groups for their introduction. However, coarse notions of hyperbolicity can be found in some of his ...
4
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1answer
302 views

When does a subgroup of $\operatorname{GL}(n, \mathbb Q)$ have a bounded fundamental domain on $\mathbb R^n$?

$\DeclareMathOperator\GL{GL}$Let $G \subset M_{n\times n~}(\mathbb Z)$ be a finitely generated subgroup of $\GL(n,\mathbb Q)$ (i.e. $g\in G$ is an invertible matrix with entries in $\mathbb Z$). Then $...
4
votes
1answer
108 views

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 ...
8
votes
1answer
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 ...
4
votes
1answer
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 ...
7
votes
1answer
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 ...
8
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0answers
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 ...
2
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2answers
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 ...
8
votes
1answer
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 ...
13
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2answers
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 ...
7
votes
1answer
122 views

Uniqueness of stationary measures for $(G,\mu)$ boundaries

Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup. Let $\nu$ is a $\...
6
votes
1answer
206 views

Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?

Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$. Question. Is the function $k(g,h) = \...
4
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0answers
110 views

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 ...
7
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0answers
105 views

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 ...
7
votes
1answer
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 $f:VX \to VY$ ($VX,\,VY$ ...
7
votes
1answer
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 ...
2
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0answers
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 $G_i$ ...
7
votes
3answers
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 ...
6
votes
0answers
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 ...
7
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2answers
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 ...
4
votes
0answers
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 ...
8
votes
2answers
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 $n \geq 3$ (...
5
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0answers
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 [1] constructed examples of geometrically infinite hyperbolic ...
3
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0answers
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 ...
3
votes
0answers
131 views

Thompson's group F and algebraic links

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 ...
9
votes
2answers
413 views

Braid groups and Kazhdan's property (T)

In Nica's dissertation Group actions on median spaces, we can read the following assertion: Braid groups do not contain infinite subgroups satisfying Kazhdan's property (T). This is used in order to ...

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