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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questions
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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
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1answer
172 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
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1answer
98 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
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1answer
185 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) = \...
3
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0answers
96 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
102 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
200 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
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1answer
115 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
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3answers
355 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
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0answers
131 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 ...
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137 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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0answers
45 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
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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
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2answers
197 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
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2answers
192 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
90 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
116 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
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0answers
118 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
387 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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0answers
82 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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0answers
79 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
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1answer
186 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
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1answer
631 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
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1answer
225 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
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3answers
201 views
Order from Coxeter-Dynkin diagram
How is the order of a Coxeter group determined from its Coxeter-Dynkin diagram?
11
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2answers
530 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
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0answers
103 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
226 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
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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
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1answer
254 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
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3answers
561 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 ...
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0answers
34 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{...
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1answer
73 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 ...
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93 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
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1answer
64 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
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0answers
48 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
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0answers
114 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
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1answer
410 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?
1
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1answer
234 views
What is the symmetry group of this compound of two polytopes?
The geometric shape in question is a compound of two polytopes: an 11-hypercube with edge length $2$ and an 11-simplex with edge length $\sqrt6$ whose vertices are a subset of the hypercube’s. What is ...
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2answers
336 views
Sequence of epimorphisms of residually finite groups stabilizes
Let $G_1 \to G_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually ...
7
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1answer
245 views
An approach to showing hyperbolic groups are CAT(0)
I've been sitting on this idea for quite a while but I'm not in academia any longer so not likely to ever tackle it on my own. The approach is as follows:
$G$ acts on its boundary $\partial G$
ergo, $...
7
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1answer
223 views
Virtually large groups of small rank (related to 3-manifolds)
I am looking for a reason why a 3-manifold group $G$ that is virtually $\mathbb{Z}\times F$, $F$ being either non-cyclic free or a surface group, does not admit a presentation on two generators.
These ...
3
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3answers
238 views
Følner sequences with weird shapes
Let $G$ be a discrete and finitely generated group. Recall that $\{F_n\}_{n \in \mathbb{N}}$ is a Følner sequence if $|g F_n \cup F_n|/|F_n| \rightarrow 1$ for every $g \in G$. As is well known, ...
4
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1answer
273 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 $...
5
votes
2answers
235 views
profinite completion and linear representations of finitely presented groups
Let $G$ be a finitely presented group. It is clear that if the profinite completion $\widehat{G} $ of $G$ is finite, then any finite dimensional complex linear representation $\rho: G\to \text{GL}(m, \...
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0answers
128 views
Groups whose exponential growth is never pinched
This question is motivated by a partial answer to another question on MO.
Given an infinite finitely generated group $G$ and a finite generating set $S$, let $b_n^S$ be the cardinality of the ball of ...
3
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0answers
72 views
An analogue of the Milnor-Švarc lemma for Busemann boundaries
The Milnor-Švarc lemma, is, without doubt, regarded as one of the most important statements in geometric group theory. (Edit) One of the corollaries of this lemma says that if a hyperbolic group $G$ ...
6
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1answer
89 views
Example of an invariant metric on a nilpotent group which is not asymptotically geodesic
Let $X$ be a metric space. We say that $X$ is asymptotically geodesic if for all $\epsilon > 0$, there exists $R > 0$ such that, for all $x,y \in X$, there exists some finite sequence of points $...
3
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0answers
137 views
A pair of non-free groups, each of which is isomorphic to the free product of itself with a free group. Are they isomorphic?
This is a follow-up question to Is there a non-free group $G$ whose subgroups are all freely decomposable?
In the answer to that question, Cornulier gives the following example (due to Kurosh) of a ...
