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
10 votes
1 answer
359 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 ...
AGenevois's user avatar
  • 7,481
16 votes
3 answers
665 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 ...
Matt Zaremsky's user avatar
1 vote
0 answers
43 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{...
24601's user avatar
  • 250
0 votes
1 answer
172 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 ...
frafour's user avatar
  • 435
0 votes
0 answers
159 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?
Daniel Sebald's user avatar
2 votes
1 answer
101 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 ...
Sangrok Oh's user avatar
2 votes
0 answers
78 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 $\mathcal{C}$ ...
JohannesPauling's user avatar
5 votes
0 answers
191 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 ...
Ethan Dlugie's user avatar
  • 1,247
9 votes
1 answer
486 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?
Daniel Sebald's user avatar
1 vote
1 answer
344 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 ...
Daniel Sebald's user avatar
10 votes
2 answers
437 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 ...
frafour's user avatar
  • 435
8 votes
1 answer
425 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, $...
BGroff's user avatar
  • 151
7 votes
1 answer
353 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 ...
lemon314's user avatar
  • 323
3 votes
3 answers
476 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, ...
Diego Martinez's user avatar
4 votes
1 answer
389 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 $...
Li Yutong's user avatar
  • 3,362
6 votes
2 answers
322 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, \...
Bruno's user avatar
  • 497
5 votes
0 answers
141 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 ...
ARG's user avatar
  • 4,342
4 votes
0 answers
170 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 $G$ ...
Peter Kosenko's user avatar
5 votes
1 answer
169 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 $...
Christian Gorski's user avatar
4 votes
0 answers
163 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 ...
seldom seen's user avatar
2 votes
0 answers
197 views

Algebraic rigidity in the automorphism group of the Cantor set

Let C be a Cantor set (middle third). Now we know that C is a totally disconnected compact topological space with the natural topology (i.e., $C=\{0,1\}^{\mathbb{N}}$). Let G:=Homeo(C) be the set of ...
Sushil's user avatar
  • 41
1 vote
0 answers
123 views

Uniform divergence of geodesics in RAAGs

$\DeclareMathOperator\div{div}$Let $(X,d)$ be a metric space and let $\gamma$ be a geodesic in $X$. Roughly speaking, the divergence of $\gamma$ at a point $x\in \gamma$ is a function $\div:\mathbb{R}...
M. Dus's user avatar
  • 1,900
4 votes
1 answer
295 views

Characterizations of groups whose general linear representations are all trivial

Let $G$ be a group. Suppose for any general linear representation $\rho:G\to\mathrm{GL}(n)$, $\rho$ must be trivial. Question: Are there any characterizations or equivalent conditions for $G$? Thanks ...
Shiquan Ren's user avatar
  • 1,970
10 votes
2 answers
519 views

Gromov hyperbolicity constant vs. Gromov-Hausdorff distance to a tree

Let $X$ be a compact, geodesic metric space which is Gromov hyperbolic with a constant $\delta>0$. To fix scaling, let us also assume that $X$ has diameter $1$. To fix a definition of Gromov ...
anon's user avatar
  • 101
19 votes
1 answer
379 views

Is there a simple group that is torsion-free, type $\textrm{F}_\infty$, and infinite dimensional?

Does there exist an example of a group that is: Simple, Torsion-free, Of type $\textrm{F}_\infty$, and Infinite dimensional (meaning of infinite cohomological dimension)? Thompson's group $F$ has ...
Matt Zaremsky's user avatar
6 votes
1 answer
548 views

Generalized Birman exact sequence for surfaces with boundaries

Let $S_g^n$ be a surface of genus g with n boundaries and let $Mod(S_g^n)$ be its mapping class group. We will also denote by $S_{g,m}^n$ a surface of genus g with n boundaries and m punctures. The ...
Philippe Tranchida's user avatar
6 votes
2 answers
518 views

Which groups are doubling?

A metric space $(M,d)$ is doubling if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric ...
Ville Salo's user avatar
  • 6,337
6 votes
1 answer
153 views

Lower bound on growth for intersection of two subgroups of free group

Let $\Gamma_1$ and $\Gamma_2$ be two subgroups of the rank-$2$ free group $F_2$. Can then one find a nontrivial lower bound on the growth exponent of their intersection $\Gamma_1 \cap \Gamma_2$, in ...
Ilia Smilga's user avatar
  • 1,364
2 votes
0 answers
193 views

Is every cocompact lattice in Sp(n,1) residually finite?

We know that every finitely generated linear group over a commutative ring is a residually finite group by Mal'cev's theorem, but each cocompact lattice in Sp(n,1) is a finitely generated linear group ...
Jianguo Zhang's user avatar
18 votes
1 answer
540 views

Is Thompson's group $T$ co-Hopfian?

A group $G$ is co-Hopfian if every injective homomorphism $G\to G$ is bijective, i.e., if $G$ contains no proper subgroups isomorphic to $G$. My question is whether Thompson's group $T$ is co-Hopfian. ...
Matt Zaremsky's user avatar
8 votes
2 answers
443 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 ...
Uzu Lim's user avatar
  • 821
5 votes
1 answer
320 views

"Simplicial complex" product of groups?

Let $X=(V,E)$ be a graph, and to each vertex $v \in V$, associate a group $G_v$. The graph product of the groups $G_v$ (as defined e.g. here) is $F/R$; the quotient of the free product of the $G_v$ by ...
Matt's user avatar
  • 198
1 vote
0 answers
87 views

Weil-Petersson metric with respect to covering

Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\...
Cusp's user avatar
  • 1,703
5 votes
2 answers
475 views

Is every countable discrete group a subgroup of a non discrete Lie group?

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group? 2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word ...
Ali Taghavi's user avatar
2 votes
0 answers
137 views

Term and theories about "relation-free" elements in a group?

For a group $G$, there are two elements a, b which are "relation-free", i.e., there is no nonempty, reduced word $W(X,Y)$ such that $W(a,b)=1$ in $G$. Is there any terminologies or theories ...
qkqh's user avatar
  • 347
7 votes
0 answers
419 views

Are these two kernels isomorphic groups?

We have a finitely presented, infinite group $\mathsf{B}$, coming from a geometric topology problem (it is the quotient of a braid group for a genus 2 surface). It is generated by elements \begin{...
Francesco Polizzi's user avatar
10 votes
0 answers
212 views

Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width. Let $X$ be a geodesically complete CAT(0) ...
Yellow Pig's user avatar
  • 2,480
11 votes
1 answer
358 views

Are groups with the Haagerup property hyperlinear?

In his 2008 paper Hyperlinear and Sofic Groups: A Brief Guide, Pestov asked (Open Question 9.5) whether every group with the Haagerup property is hyperlinear (or sofic). Has this question been ...
MaoWao's user avatar
  • 1,027
9 votes
0 answers
350 views

Infinite-dimensional torsion-free $F_\infty$-group not containing $F$

Is there an example of a group $G$ that has the properties the cohomological dimension of $G$ is infinite: $\operatorname{cd}(G) = \infty$, $G$ is torsion-free, $G$ is of type $F_\infty$, $G$ does ...
Stefan Witzel's user avatar
3 votes
1 answer
229 views

Geometric content of area of a word in geometric group theory?

Where does the idea of 'area' come from in Geometric Group Theory? The wikipedia article states that this definition was 'inspired' from Riemannian geometry: Gromov's proof was in large part informed ...
Siddharth Bhat's user avatar
3 votes
0 answers
106 views

Are all intermediate growth branch groups just-infinite?

Are all branch groups of intermediate growth just-infinite? I can't seem to find an answer to this one way or another; the question is motivated by the fact all examples of intermediate growth branch ...
Thomas Meyer's user avatar
7 votes
3 answers
1k views

Bass-Serre theory textbook

I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? ...
George K's user avatar
  • 422
8 votes
0 answers
141 views

Small flag triangulations

In geometric group theory and low-dimensional topology, asking for a triangulation of a specific space to be flag can often be somewhat more cumbersome than just turning a CW-complex into a simplicial ...
vladkvankov's user avatar
1 vote
0 answers
110 views

Question on models for $EG$ for a $G$-CW complex

I am having trouble finding information on a definition in P. Hanham's PhD thesis paper. recall that given a discrete group $G$ a $G$-CW-complex $X$ is a CW-complex equipped with a topological $G$ ...
Dominic Petti's user avatar
4 votes
0 answers
146 views

clarification on the proof of Bestvina-Mess formula

I am studying Bestvina and Mess's results on the boundary of hyperbolic groups [The Boundary of Negatively Curved Groups. Journal of the American Mathematical Society Vol. 4, No. 3 (Jul., 1991), pp. ...
Brandy's user avatar
  • 41
2 votes
0 answers
186 views

$V$-like actions of $V$

This continues my question about prefix-continuous bijections (since the answer was "yes"). Notation and conventions: Let $A$ be a finite alphabet and $L \subset A^*$ a language. Let $G$ be a group. ...
Ville Salo's user avatar
  • 6,337
6 votes
1 answer
186 views

Is there a prefix-continuous bijection between finite words and eventually zero words?

Let $$ X = \{x \in \{0,1\}^{\omega} \;|\; \exists m: \forall i \geq m: x_i = 0\} $$ (one-way infinite eventually zero words). Let $\{0,1\}^*$ denote the finite (not necessarily nonempty) words over $\{...
Ville Salo's user avatar
  • 6,337
10 votes
1 answer
531 views

hyperbolic quotient of hyperbolic group

I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...
Derek Holt's user avatar
  • 36.4k
0 votes
0 answers
79 views

Large gaps in the norm of a subgroup and its centraliser

Take an infinite finitely generated group $G$ with an infinite subgroup $N$ which has an infinite centraliser $Z = Z_G(N)$. Let $S$ be some [symmetric] generating set of $G$ and for $g \in G$, ...
ARG's user avatar
  • 4,342
4 votes
0 answers
205 views

Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
qkqh's user avatar
  • 347

1
4 5
6
7 8
19