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
943
questions
10
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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
3
answers
665
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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
0
answers
43
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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
1
answer
172
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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
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0
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159
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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
1
answer
101
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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
0
answers
78
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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
0
answers
191
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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 ...
9
votes
1
answer
486
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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
vote
1
answer
344
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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 ...
10
votes
2
answers
437
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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 ...
8
votes
1
answer
425
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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
votes
1
answer
353
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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 ...
3
votes
3
answers
476
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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
votes
1
answer
389
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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 $...
6
votes
2
answers
322
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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, \...
5
votes
0
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141
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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 ...
4
votes
0
answers
170
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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 $G$ ...
5
votes
1
answer
169
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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 $...
4
votes
0
answers
163
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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 ...
2
votes
0
answers
197
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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 ...
1
vote
0
answers
123
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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}...
4
votes
1
answer
295
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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 ...
10
votes
2
answers
519
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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 ...
19
votes
1
answer
379
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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 ...
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 ...
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 ...
6
votes
1
answer
153
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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 ...
2
votes
0
answers
193
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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 ...
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.
...
8
votes
2
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443
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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 ...
5
votes
1
answer
320
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"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 ...
1
vote
0
answers
87
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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\...
5
votes
2
answers
475
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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 ...
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 ...
7
votes
0
answers
419
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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{...
10
votes
0
answers
212
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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) ...
11
votes
1
answer
358
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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 ...
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 ...
3
votes
1
answer
229
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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 ...
3
votes
0
answers
106
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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 ...
7
votes
3
answers
1k
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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? ...
8
votes
0
answers
141
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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 ...
1
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0
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110
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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 $G$ ...
4
votes
0
answers
146
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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. ...
2
votes
0
answers
186
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$V$-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. ...
6
votes
1
answer
186
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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 $\{...
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^...
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 $g \in G$,
...
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 ...