# 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

746
questions

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votes

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63 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 ...

**8**

votes

**0**answers

95 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**

votes

**0**answers

103 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 \...

**4**

votes

**1**answer

154 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 ...

**7**

votes

**2**answers

517 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 ...

**14**

votes

**1**answer

277 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 $$M//G = M \times_G EG$$...

**4**

votes

**1**answer

100 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 ...

**6**

votes

**1**answer

172 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 $\mathcal{F}$...

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vote

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87 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**

votes

**1**answer

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**

votes

**0**answers

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;
$\bullet$ ...

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votes

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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**

votes

**3**answers

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 ...

**10**

votes

**2**answers

324 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 ...

**3**

votes

**0**answers

80 views

### Examples of nonlinear residually finite hyperbolic groups

What are some examples of nonlinear residually finite hyperbolic groups?

**29**

votes

**16**answers

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 ...

**2**

votes

**0**answers

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**

votes

**1**answer

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 ...

**14**

votes

**2**answers

575 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**

votes

**1**answer

136 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 ...

**8**

votes

**0**answers

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 ...

**8**

votes

**1**answer

246 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 ...

**2**

votes

**2**answers

208 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

**1**answer

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 ...

**7**

votes

**1**answer

120 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

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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 ...

**7**

votes

**3**answers

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

**0**answers

141 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**

votes

**1**answer

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 ...

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votes

**0**answers

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 $G_i$ ...

**4**

votes

**0**answers

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 ...

**7**

votes

**2**answers

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 ...

**8**

votes

**2**answers

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 $n \geq 3$ (...

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votes

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**2**answers

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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vote

**0**answers

86 views

### Non-discrete subgroups acting on Euclidean spaces

I'm curious about finitely generated subgroups in the isometry group of Euclidean spaces. I know the isometry group is the semi-direct product of translation groups with orthogonal groups. Also ...

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votes

**0**answers

83 views

### Intersection of descending series in a free group

I have stumbled upon a problem. It can be stated in the following way: Let $E$ be a finitely generated free group. Denote $\gamma_n(E)$ the $n$-th term of the lower central series. Consider a ...

**4**

votes

**1**answer

190 views

### Subgroups of $W(E_8)$

Are there any proper subgroups of the Coxeter group $W(E_8)$ which are also proper overgroups of $W(A_8)$, other than $\text{Aut}(A_8)$?

**17**

votes

**1**answer

649 views

### Are there any "simple" monoids with intermediate growth?

The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...

**9**

votes

**1**answer

232 views

### Largest Hopfian quotient

Let $\Gamma$ be a group, say finitely generated if it helps. Does $\Gamma$ admit a largest Hopfian quotient? That is, does there exist a Hopfian quotient $H$ of $\Gamma$, such that every surjective ...

**5**

votes

**3**answers

222 views

### Order from Coxeter-Dynkin diagram

How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?

**11**

votes

**2**answers

539 views

### Constraints on the homology of amenable groups

Edit (March 24): My first question has been answered nicely, but I am still looking for an answer to the second one.
Due to the Kan–Thurston theorem, the homology of an arbitrary group can be anything ...

**9**

votes

**0**answers

107 views

### Artin groups of type $D_n$ as mapping class groups?

According to Allcock (Braid Pictures for Artin groups, https://arxiv.org/abs/math/9907194), the Artin group $A(D_n$) of type $D_n$ may be realized as an index 2 subgroup of the orbifold fundamental ...

**8**

votes

**1**answer

235 views

### Cohomological dimension bounds on the fundamental group of a manifold

Suppose $M$ is a (closed, connected, oriented, smooth) manifold.
If $M$ is aspherical, i.e., if the inversal covering $\tilde{M}$ is contractible, $M$ is a $B\pi_1(M)$. This is often enforced by ...