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
948
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When the fundamental group of subgraph of groups embeds?
Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
4
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0
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59
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On the induction step in Theorem 2.6 of "Homological stability for linear groups" by Kallen
I am currently reading the proof of the connectivity theorem of Wilberd van der Kallen (Theorem 2.6 in https://link.springer.com/article/10.1007/BF01390018) for a seminar talk. I am a little stuck on ...
6
votes
1
answer
149
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
8
votes
4
answers
498
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Residual finiteness of hyperbolic 3-manifold groups
So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is:
Q1. If $M$ is an infinite-...
4
votes
1
answer
197
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Injective hulls of metric spaces
In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
3
votes
1
answer
144
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Green's kernel estimates on finitely generated groups
I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...
2
votes
0
answers
226
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Interpretation of Kazhdan T property cohomologically
$\newcommand{\triv}{\mathit{triv}}$A group $G$ has property $T$ if $\triv$ is isolated in the space of unitary representations with the Fell topology.
In general, we heuristically have $H^1(G,Ad(V))$ (...
6
votes
1
answer
360
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Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
0
votes
1
answer
100
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Finding automorphism groups of regular graphs [closed]
Can some body help me with some source code for finding automorphism groups of regular maps?. For example: the type of graph is denoted as $\{p, q\}$, which means that they are tessellations of the ...
3
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0
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135
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A question about Gromov's proof of a "more effective version of the main theorem"
In the paper "Groups of polynomial growth and expanding maps" Gromov proves the following "effective version of the main theorem"
For any positive integers $d$ and $k$, there ...
4
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0
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384
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Problem 1.8 from Kirby's list
Context
I looked through a book called "Problems in Low-Dimensional Topology", where Rob Kirby lists a set of problems. He provides a list of problems, states their conjectures, and ...
0
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148
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Residual finiteness of semidirect product $\mathbb{Z}^2\ltimes \mathbb{Z}[1/10]$ of abelian groups
Let $\mathbb{Z}[1/10]$ be an abelian group by addition. Let $\mathbb{Z}^2$ act on it by automorphisms by $x\mapsto 2x$ and $x\mapsto 5x$. Is the corresponding semidirect product $\mathbb{Z}^2\ltimes \...
7
votes
1
answer
249
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Relation between Floyd and Gromov boundaries of hyperbolic groups
Let $G$ be a hyperbolic group. Consider the Floyd boundary as defined in https://www.unige.ch/math/folks/karlsson/free.pdf by Karlsson. For a Floyd function $f$, we denote the Floyd boundary of $G$ by ...
5
votes
1
answer
239
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Word length in the surface groups
I want to know if there are some results about the title of this question.
Let $G$ be an orientable closed surface group with genus $n$ greater than 1. We know it has a canonical presentation.
$$G=\...
9
votes
3
answers
476
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Residually solvable Bianchi groups
Let $d$ be a square-free positive integer, and let $\mathcal{O}_d$ be the ring of integers of the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$. Consider the Bianchi group $\Gamma_d = \...
1
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0
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80
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$L^p$-compression of metabelian groups
Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
5
votes
0
answers
189
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Virtual fibring of $\mathrm{Out}(F_2\times F_2)$
A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated.
I want ...
23
votes
1
answer
1k
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Universal graph
A connected (and infinite) graph $U$ will be called $n$-universal if any connected graph with degree $\leqslant n$ admits an embedding in $U$.
Is there a 3-universal graph with bounded degree?
0
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0
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70
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Lifting of algebraic fibrations via a subnormal series
Consider a finitely generated group $G$ and a subnormal series of $G$:
$$1=G_0\trianglelefteq G_1\trianglelefteq\cdots\trianglelefteq G_{n-1}\trianglelefteq G_n=G$$
Now, suppose that $G_1$ fibres, i.e....
2
votes
2
answers
287
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When is $\smash{\check{H}}^{q}(X,A;R)\cong H_{c}^{q}(X-A;R)$ for a pair $(X,A)$?
I'm trying to understand the proof of Corollary 1.3 part b. in a paper by Bestvina and Mess titled 'The Boundary of negatively curved groups'. I do not understand why $\smash{\check{H}}^{q}(X,A;R)\...
12
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2
answers
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Group generated by two irrational plane rotations
What groups can arise as being generated by two rotations in $\mathbb R^2$ by angles $\not \in \mathbb Q\pi$?
If the centers of the rotations coincide, then the rotations commute and generate some ...
1
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0
answers
145
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Does every amenable group $G$ admit a two-sided Folner sequence?
By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence.
Context: I just came up with this question and surprisingly I haven'...
12
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3
answers
943
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Road map to learn about $\operatorname{Out}{F_n}$
I'm a last year undergraduate student and I have taken a graduate course in geometric group theory.
I'd like to start reading some more advanced stuff in geometric group theory and in particular about ...
6
votes
1
answer
142
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Translation length on annular curve graphs
Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked.
Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...
2
votes
1
answer
155
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Commensurability classes of subgroups of a nilpotent group
Here is a question I stumbled upon in my research.
Question: Given a finitely generated nilpotent group $G$, do its subgroups fall into finitely many abstract commensurability classes?
Recall that two ...
0
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0
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149
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The geometric models of generalised Baumslag-Solitar groups
I am trying to understand a construction in the paper "The large scale geometry of the higher Baumslag-Solitar groups", GAFA, Geometric and functional analysis 11, 1327–1343 (2001), ...
19
votes
1
answer
381
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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 ...
2
votes
1
answer
165
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On the existence, for $\langle X,R\rangle$ a finite presentation of a group $G$, of an exact sequence of $\mathbb{Z}G$ modules
From this Q&A -- for $\langle X,R\rangle$ a finite presentation of a group $G$, there is an exact sequence of $\mathbb{Z}G$ modules
$$0\rightarrow\pi_{2}(Z)\rightarrow \mathbb{Z}G^{\oplus R}\...
3
votes
0
answers
217
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What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?
Follow up question, edited in on 12/20 below:
Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
4
votes
0
answers
171
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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 $G$ ...
4
votes
1
answer
197
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Reference for Chebyshev centers
Today, I came across the concept of Chebyshev center twice.
In particular, it is the key tool in the very elegant paper "A fixed point theorem for $L^1$ spaces" by Bader, Gelander and Monod.
...
1
vote
1
answer
119
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Divergence functions in hyperbolic groups
Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.
We note that in $\mathbb{R}^2$ there is no divergence ...
0
votes
0
answers
83
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Distances on spheres in Cayley graphs of non-amenable groups
Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...
7
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0
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97
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Normal subgroups of pure braid groups stable under strand bifurcation
$\DeclareMathOperator\PB{PB}\DeclareMathOperator\B{B}$Let $\PB_n$ be the $n$-strand pure braid group. For each $1\le k\le n$, let $\kappa_k^n \colon \PB_n \to \PB_{n+1}$ be the monomorphism that takes ...
5
votes
1
answer
123
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Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
14
votes
1
answer
982
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Recognizing free groups
While I’m aware (and as it was pointed out in the comments) it is generally undecidable whether a given presentation represents a free group, I’m interested in criteria that nevertheless ensure that ...
0
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0
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164
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Research directions related to the Hilbert-Smith conjecture
The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
2
votes
1
answer
236
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Do balls in expander graphs have small expansion?
Consider a $d$-regular infinite transitive expander graph $G$, and let $B_r$ be a ball of radius $r$ in $G$. Can one place any upper bounds on the expansion of $B_r$?
My intuition is that $B_r$ will ...
2
votes
0
answers
96
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Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
4
votes
4
answers
288
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Groups acting non-properly cocompactly on hyperbolic spaces
A group $G$ is hyperbolic if it admits a geometric (the action is proper and co-bounded) action on a geodesic hyperbolic metric space. Also, the definition can be given as follows, a group $G$ ...
4
votes
1
answer
142
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Salvetti complex of dihedral Artin group
The Salvetti complex of a RAAG is well-known and it is fairly simple, since each complete graph gives rise to a tori. The case of Artin groups is wilder, since we do not have tori anymore. The ...
7
votes
1
answer
430
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Groups acting on infinite dimensional CAT(0) cube complex
I have seen many examples where a finitely generated infinite group acts properly/freely by isometry on finite dimensional CAT(0) cube complexes. Examples of such groups are discussed in many articles....
7
votes
2
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438
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Amalgamated product acting on CAT(0) cube complex
I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...
5
votes
0
answers
202
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What is known about the upper density of torsion elements in finitely generated groups?
Let $T\subset G$ be the set of all torsion elements in a finitely generated infinite group $G$, and let $B_n\subset G$ be the closed ball of radius $n$ around $1$ w.r.t. to the word metric for some ...
2
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0
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129
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Need help understanding the geometry of a particular building structure
$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...
1
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0
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126
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A generalisation of residual finiteness?
A group $\Gamma$ is Residually Finite (RF) if
$\forall g \neq e \in \Gamma$ there is a homomorphism $h: \Gamma \to G$ where $G$ is a finite group such that $h(g) \neq e$. Free groups are known to be ...
3
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0
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92
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Relation of geometric and polyhedral convergence
By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
1
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0
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112
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Help to understand the geodesics in $BS(1, 2)$
I would like to understand the sets of geodesics in $BS(1, 2)$, which is described in https://arxiv.org/pdf/1908.05321.pdf, Proposition 3 (page 3).
Write $$ G=B S(1, 2)=\left\langle a, t \mid t a
t^{...
3
votes
0
answers
291
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Is G(4,7) a Coxeter group
Let $G(4, 7)$ be an abstract group with the presentation
$$\langle a,b,c | a^2 = b^2 = c^2 = 1, (ab)^4 = (bc)^4 = (ca)^4 = 1, (acbc)^7 = (baca)^7 = (cbab)^7 = 1 \rangle $$
Richard Schwartz considered ...
2
votes
0
answers
120
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Further questions to limit groups and an article of Fujiwara and Sela
I already have asked a question to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, ...