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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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 $H^1(G,Ad(V))$ (...
user135743's user avatar
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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 ...
A Name's user avatar
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6 votes
1 answer
355 views

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 ...
Denis T's user avatar
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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 ...
Zahid Malik's user avatar
4 votes
0 answers
377 views

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 ...
Amanuel Jissa's user avatar
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147 views

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 \...
Anna Hendrix's user avatar
5 votes
1 answer
236 views

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=\...
keqiyehuopo's user avatar
5 votes
0 answers
188 views

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 ...
Marcos's user avatar
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23 votes
1 answer
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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?
Anton Petrunin's user avatar
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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....
Marcos's user avatar
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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 ...
ARG's user avatar
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7 votes
1 answer
245 views

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 ...
Subhajit Chakraborty's user avatar
12 votes
2 answers
1k views

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 ...
Ethan Dlugie's user avatar
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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'...
Saúl RM's user avatar
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2 votes
2 answers
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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)\...
Harsh Patil's user avatar
9 votes
3 answers
475 views

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 = \...
Carl-Fredrik Nyberg Brodda's user avatar
2 votes
1 answer
155 views

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 ...
Corentin B's user avatar
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0 answers
148 views

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), ...
JB T's user avatar
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2 votes
1 answer
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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}\...
Souvik Mandal 's user avatar
3 votes
0 answers
217 views

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 ...
Chase's user avatar
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1 vote
1 answer
116 views

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 ...
Strichcoder's user avatar
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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 ...
user3521569's user avatar
7 votes
0 answers
97 views

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 ...
Matt Zaremsky's user avatar
5 votes
1 answer
122 views

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 $\...
Takao Hishikori's user avatar
14 votes
1 answer
980 views

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 ...
ThorbenK's user avatar
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2 votes
0 answers
95 views

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 ...
ggt001's user avatar
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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 $A_p$ ...
sadman-ncc's user avatar
2 votes
1 answer
234 views

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 ...
user3521569's user avatar
4 votes
1 answer
134 views

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 ...
Marcos's user avatar
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4 votes
4 answers
283 views

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 $G$ ...
bishop1989's user avatar
3 votes
1 answer
150 views

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 ...
Sebastian's user avatar
7 votes
1 answer
424 views

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....
bishop1989's user avatar
7 votes
2 answers
436 views

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 $...
bishop1989's user avatar
5 votes
0 answers
201 views

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 ...
I. Haage's user avatar
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2 votes
0 answers
128 views

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 ...
jsch's user avatar
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1 vote
0 answers
125 views

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 ...
mathstudent42's user avatar
1 vote
0 answers
112 views

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^{...
ghc1997's user avatar
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3 votes
0 answers
92 views

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 ...
bergfalk's user avatar
3 votes
0 answers
290 views

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 ...
Shijie Gu's user avatar
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2 votes
0 answers
57 views

upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups

Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
ghc1997's user avatar
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8 votes
1 answer
671 views

Where to find English translation of Pansu's paper from Ann. Math?

Where can I find English translation of the following paper? P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...
Piotr Hajlasz's user avatar
2 votes
0 answers
146 views

The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$

Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...
ghc1997's user avatar
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1 vote
0 answers
153 views

Solution of an equation over free group

Let $F_n$ be a free group on $n$ generators. Let $w \in F_n$ be a word such that there does not exist any solution in $F_n$ for the equation $w.w(t_1, \ldots, t_n) = 1$, where $t_1, \ldots, t_n$ are ...
Shri's user avatar
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2 votes
0 answers
132 views

Strong converse of Kazhdan's property (T)

In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...
MaoWao's user avatar
  • 1,027
6 votes
1 answer
237 views

Does the inner automorphism group of the fundamental group of a closed aspherical manifold always have an element of infinite order?

Let $\pi_1$ be the fundamental group of a closed aspherical manifold of dimension $n$. In particular, $\pi_1$ is finitely presented, torsion-free and its cohomology is finitely generated and satisfies ...
user513804's user avatar
1 vote
0 answers
72 views

Automorphic images of cones in free group

Let $F_2$ be the free group with basis $\{a,b\}$, with corresponding word metric $d$. For $x\in F_2$, the cone $C(x)$ is $C(x):=\{y\in F_2\mid d(1,y)=d(1,x)+d(x,y)\}$, that is, the set of elements ...
Matt Zaremsky's user avatar
1 vote
0 answers
72 views

Basis of subgroup of free group

Let $F_2$ be a free group on $2$ generators $a, b$. We know $b$ and a conjugate of $b$, which is different from $b$, generate rank 2 free subgroup of $F_2$ and they are free generating set of the ...
Infy's user avatar
  • 11
0 votes
0 answers
125 views

Examples of a group with infinitely many ends which are not represented as a free product of groups

Let $F_1$ and $F_2$-non-trivial groups. Is it correct that the number of ends of the free product $F_1\ast F_2$ is infinite? My thoughts about this: Since $e(G)=\infty$ then $G=F_1\ast F_2$, a non-...
UserIn's user avatar
  • 103
2 votes
1 answer
178 views

Markov property for groups?

My question again refers 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, arXiv:...
TheMathematician's user avatar
1 vote
0 answers
130 views

The free products of finitely many finitely generated groups are hyperbolic relative to the factors

Are there any references how to show that:the free products of finitely many finitely generated groups are hyperbolic relative to the free factors. More precisely, how to show that $G = A \ast B $ is ...
Kalye's user avatar
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