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

Filter by
Sorted by
Tagged with
8
votes
2answers
205 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 ...
5
votes
0answers
79 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 ...
9
votes
1answer
190 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
0answers
72 views

Examples of nonlinear residually finite hyperbolic groups

What are some examples of nonlinear residually finite hyperbolic groups?
26
votes
13answers
1k 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 ...
7
votes
2answers
486 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
votes
0answers
68 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
2answers
269 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
votes
1answer
262 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 ...
13
votes
2answers
554 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
1answer
296 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
105 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
243 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
127 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
314 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
votes
0answers
105 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
votes
2answers
200 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
190 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
votes
2answers
482 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
116 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
200 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
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
votes
0answers
103 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
208 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
117 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
votes
0answers
47 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
364 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
138 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
2answers
207 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
59 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
201 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
votes
0answers
99 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
0answers
119 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
124 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
408 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 ...
1
vote
0answers
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 ...
0
votes
0answers
82 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
1answer
187 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
1answer
642 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 ...
11
votes
2answers
532 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 ...
18
votes
4answers
2k views

Units in the group ring over fours group after Gardam

Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$. This conjecture asserted that if $\Gamma$ is a ...
5
votes
3answers
214 views

Order from Coxeter-Dynkin diagram

How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?
7
votes
3answers
595 views

Is there a one relator group with property (T)?

Is there a one-relator group with property (T)? That is, is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup ...
9
votes
1answer
231 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 ...
21
votes
4answers
1k views

Free splittings of one-relator groups

Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is it ...
9
votes
0answers
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
1answer
229 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 ...
6
votes
0answers
80 views

Indices of Coxeter groups in themselves

Every Euclidean Coxeter group ($P_n$, $Q_n$, $R_n$, $S_n$, $T_7$, $T_8$, $T_9$, $U_5$, $V_3$, $W_2$) contains infinitely many scaled copies of itself as subgroups. What are all the possible indices of ...
11
votes
1answer
263 views

Translation lengths in CAT(0) spaces

Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...
16
votes
3answers
566 views

Group with non-trivial center containing trivially-intersecting copies of itself

I'm trying to think of an example of a group $G$ with non-trivial center such that there exist subgroups $H_1,H_2\le G$ both isomorphic to $G$ and satisfying $H_1\cap H_2=\{1\}$. Does such a group ...

1
2 3 4 5
15