# 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

973
questions

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### Polynomial isoperimetric inequalities for finitely presented subdirect products of limit groups

Does every finitely presented subdirect product of limit groups admit a polynomial isoperimetric inequality? That is, does there exist a constant $C > 0$ and an integer $d \geq 1$ such that for ...

6
votes

2
answers

902
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### Groups killed by centralizing one element

What groups $G$ contains an element $g$ such that $G/(g\text{ is made central})=1$ (or equivalently $[g,G]=G$, where $[g,G]:=\langle[g,h]\colon h\in G\rangle$)?
A necessary condition is that $G$ is a ...

4
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1
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### Groups (not?) quasi-retracting onto $\mathbb{Z}$ via closest points projection

Inspired by this question we ask:
Suppose that $G$ is an infinite group. Suppose that $X$ is a finite generating set of $G$. Let $\Gamma = \Gamma(G, X)$ be the resulting Cayley graph. Does $\Gamma$ ...

5
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1
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### Groups (not) quasi-retracting onto $\mathbb{Z}$

Let $X$ and $Y$ be metric spaces, with metrics $d_X$ and $d_Y$ respectively. A function $f\colon X\to Y$ is coarse Lipschitz if there exist constants $C,D>0$ such that $d_Y(f(x),f(x'))\le C d_X(x,x'...

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### Embedding f.g. groups in 2-generated groups

Let $G$ be a finitely generated group. Can $G$ be embedded as a finite-index subgroup of a 2-generated group? 100-generated?
I strongly doubt it but I don't know a counterexample.

10
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1
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294
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### What is the minimal genus of a surface acted on by the symmetric group $S_n$?

For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...

3
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1
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134
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### $C'(\lambda)$ small cancellation for $\lambda < \frac{1}{6}$

I'm working with Gromov's density model of random groups, and a nice fact is that for a fixed density parameter $0 \leq d \leq 1$, a generic group in the density model satisfies the $C'(2d)$ small ...

2
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1
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192
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### Strong Liouville property of virtually abelian groups

Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...

6
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1
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### Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?

This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?

3
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0
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### Additive characters from coarse quotient maps

Let's consider a (finitely generated) group $\Gamma$ and a
coarse quotient map
$q\colon\Gamma\to\mathbb{R}$.
I'm interested in the 1-cocycle
$\sigma\colon\Gamma\to\ell_\infty\Gamma$,
defined by $\...

3
votes

2
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460
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### How fast does the number of "fixed" points grow compared to the size of the ball in the following group?

I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb ...

6
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1
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131
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### If $X$ is a hyperbolic, locally finite graph with $\partial X \cong S^1$, and $G$ acts cocompactly but not properly on $X$, what can we say?

It is an important and deep fact of geometric group theory that if the Gromov boundary of a hyperbolic group $G$ is a circle, then $G$ is virtually Fuchsian [Tukia, Gabai, Casson-Jungreis...]. I am ...

3
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0
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### References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections

A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...

11
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1
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227
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### Recognising the elements of the Grigorchuk group

The Grigorchuk group $\mathfrak{G}= \langle a,b,c,d \rangle$ is a group of automorphisms of the infinite rooted binary tree $\mathcal{T}_2$. Every element of $\mathfrak{G}$ can be represented by a ...

3
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153
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### A few points of clarification on the Martin boundary

Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...

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### Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...

7
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2
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### Finite normal subgroup of mapping class group

Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\...

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133
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### Model-theoretic construction of Gromov boundaries on groups

For context, I'm only a second year undergraduate mathematician, so I won't know much.
For third year, I'm hoping to do a research project. I met up with a professor who might be my supervisor today, ...

6
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1
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466
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### Are Artin-Tits groups ordered groups?

We consider Artin-Tits groups of two generators $(I_2(n))$. Are these groups ordered groups?

3
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### Measures with superexponential moments on finitely generated groups

Let $\Gamma$ be an infinite finitely generated group and let $\nu$ be a measure on $\Gamma$ which generates a transient random walk. I was reading this paper, and the authors prove many of their ...

3
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1
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### Point stabilizers of the Floyd boundary of a group

Let $G$ be a finitely generated group. Consider the Floyd boundary as defined in https://www.unige.ch/math/folks/karlsson/free.pdf by A. Karlsson. For a Floyd function f, we denote the Floyd boundary ...

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### A stronger(?) notion than uniform contractibility

Let's call a metric space $ X $ strongly contractible if there exists a
function $ \rho : \mathbb{R}_+ \to \mathbb{R}_+ $ such that for every ball $
B(x;r) $ around a point $ x \in X $ we have:
$ B(x;...

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0
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### When $G$ is amenable is true that for a set $A \in p$ which is left piecewise syndetic then $\{ x: Ax^{-1} \in p \}$ is both sided syndetic?

Let be $G$ be a discrete group. I recall the definition of syndetic sets, thick sets and piecewise syndetic sets.
Definitions:
A set $A$ is left syndetic if there exist a finite $H \subset G$ such ...

6
votes

1
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193
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### On intersection of finite index fully invariant subgroup

Let $G$ be a group. A subgroup $H$ of $G$ is said to be fully invariant if for every endomorphism $\phi $ of $G$, we have $\phi(H) \subseteq H$. For a finitely generated residually finite group $G$, ...

2
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### Amplification argument for hyperlinear groups

Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...

2
votes

1
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93
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### Fundamental domain for the action on curve complex

Suppose $S$ is a surface and the Mapping Class Group, $Mod(S)$, of $S$, is the group of self-homeomorphisms of $S$, up to isotopy. This group acts on a graph called "curve graph", denote by $...

3
votes

1
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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 ...

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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
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1
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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 ...

4
votes

1
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189
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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 ...

8
votes

4
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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-...

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0
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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 $H^1(G,Ad(V))$ (...

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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 ...

6
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1
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389
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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
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1
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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 ...

4
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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 ...

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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 \...

5
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1
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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=\...

5
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0
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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 ...

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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?

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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....

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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 ...

7
votes

1
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292
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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 ...

12
votes

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
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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'...

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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)\...

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votes

3
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485
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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 = \...

2
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1
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189
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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 ...

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163
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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), ...

2
votes

1
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172
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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}\...