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
735
questions
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 ...
8
votes
2answers
203 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 ...
9
votes
1answer
189 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 ...
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
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 ...
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
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 ...
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 ...
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 ...
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 ...
7
votes
1answer
115 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
199 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
207 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 ...
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
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 ...
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 $G_i$ ...
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 ...
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 ...
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
98 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
407 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 ...
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 ...
5
votes
3answers
214 views
Order from Coxeter-Dynkin diagram
How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?
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 ...
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 ...
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 ...
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
vote
0answers
35 views
When does bottom stratum of relative train track map give rise to irreducible outer automorphism of free groups
Let $F_n$ be the free group of finite rank $n$ and let $\mathcal{O} \in \text{Out}(F_n)$. Let $\Gamma$ be a finite graph and $f : \Gamma \to \Gamma$ a relative train track representative of $\mathcal{...
0
votes
1answer
80 views
Examples of infinitely presented non-LEF groups
A group is LEF (locally embeddable in the class of finite groups) if it embeds into an ultraproduct of finite groups. Residually finite groups are LEF and finitely presented LEF groups are residually ...
0
votes
0answers
95 views
Isomorphic Coxeter groups
After enumerating the spherical Coxeter groups, it is easy to see that no two distinct cases are isomorphic. Does the same hold for Euclidean and hyperbolic Coxeter groups?
2
votes
1answer
66 views
Weakly relatively hyperbolicity and asymptotic cone
Drutu, Sapir, Osin showed that
a finitely generated group $G$ is strongly hyperbolic relative to a finite collection $\mathcal{H}$ of subgroups if and only if any asymptotic cone is tree-graded with ...
2
votes
0answers
50 views
A quasi-isometric embedding of a convex cocompact subgroup
I am currently reading a paper where they state the following claim:
"For a convex cocompact representation $\rho: \Gamma \to G$, the existence of a cocompact invariant convex set $\mathcal{C}$ ...
5
votes
0answers
118 views
Tools for computing from group presentations
What are some tools -- either theoretical/by hand or algorithmic/by computer -- that are useful for doing computations in finitely presented groups?
In my particular case, I'm working with a finitely ...
8
votes
1answer
412 views
Can $E_8$ be enlarged?
Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
