# 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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### 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 ...

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### 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(...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...

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### 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:...

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### 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 ...

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### Analogous results in geometric group theory and Riemannian geometry?

As you can see from my other question I concern mmyself with the following article at the moment:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–...

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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, ...

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### Question to limit groups (over free groups)

My question refers to the following article (to page 26: proof of Theorem 4.1):
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10....

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### Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence?

I have a question that is related to the topic of limit groups:
Let $G$ and $H$ be finitely generated groups and let $(\varphi_n: G \to H)_{n \in \mathbb{N}}$ be a sequence of group epimorphisms. Does ...

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### Proof of the connection of the growth functions of a residually finite group and all of its finite quotients

I was reading the research article entitled "Asymptotic growth of finite groups" by Sarah Black. Professor Black makes the following statement at the bottom of page 406:
Indeed, given a f.g....

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### Is the canonical map from isometry group of a Gromov hyperbolic space to homeomorphisms of its Gromov boundary injective?

Suppose X is a proper Gromov hyperbolic space and $\partial X$ is its Gromov boundary. It is well-known that there is a canonical group homomorphism $\Phi$ from the isometry group of X to the group ...

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### Classes of groups with polynomial time isomorphism problem

It is known that the isomorphism problem for finitely presented groups is in general undecidable. What are some classes of groups whose isomorphism problem is known to be solvable in polynomial time? (...

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### A "simpler" description of the automorphism group of the lamplighter group

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references.
The lamplighter group is defined by the ...

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### Morse theory on outer space via the lengths of finitely many conjugacy classes

Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...

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### Proof of Zimmer's cocycle super-rigidity theorem

I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ...

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### Are Gromov-hyperbolic groups roughly starlike? [duplicate]

Given a Cayley graph of a finitely generated Gromov-hyperbolic group $G$, does there exists $R>0$ such that every element $g \in G$ is at most distance $R$ away from a geodesic ray starting at ...

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### Can the stabiliser of a 'parabolic end' of a group stabilise an invariant line?

Let $G$ be a group acting freely and cocompactly on an infinite-ended graph $\Gamma$. In particular, $G$ is finitely generated and acts as a convergence group on the Cantor set $\rm Ends(\Gamma)$.
Let ...

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### Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?

$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...

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### Integral over quotient of discrete group

Let $Y$ be a proper metric space. By a lattice we mean a discontinuous group of isometries $\Gamma$ with compact quotient $Y/\Gamma$. You may also assume that $\Gamma$ acts freely. Suppose we are ...

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### Commuting conjugate elements in torsion-free groups

I have come across the following question while studying projective modules over integral groups rings of torsion-free groups.
Given a non-unit $x\in G$ a torsion-free group, does there exist $g\in G$ ...

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### Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...

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### $K(\pi,1)$-conjecture ofr Artin groups behave well with respect to special subgroups. Reference-Request

For a proof for an article I would need the following result:
If $A_\Gamma$ is an Artin group such that the $K(\pi,1)$-conjecture holds for it and $\Gamma'\subset\Gamma$ is an induced subgraph, then ...

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### Examples of groups that are unknown to be acylindrically hyperbolic

Let $G$ be a group. We say that $G$ is acylindrically hyperbolic (for short, AH) if $G$ admits an isometric, acylindrical, and non-elementary action on some Gromov hyperbolic space $X$.
Here is the ...

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### Extreme amenability of topological groups and invariant means

Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...

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### Cancellation of elements in the Gromov boundary of a free group

Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...

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### Write an Artin group as an HNN-extension

Assume that $A_\Gamma$ is an Artin group and $\chi:A_\Gamma\to(\mathbb{Z},+)$ is a group homomorphism of the following form. $\Gamma=\Gamma_1\cup\Gamma_2$ with $\Gamma_1\cap\Gamma_2=\emptyset,A_{\...

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### An application of ping-pong lemma

Let $F_2$ be free group of rank two with generators $a$ and $b$. If $H$ is a subgroup of $F_2$ generated by $d\geq 2$ elements with $$H=\langle a,b^{-k}ab^k, k=1,2,...,d-1\rangle,$$ I was trying to ...

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### Barycenter maps that are "submultiplicative" with respect to group actions

Background and notation
For a set $X$, we denote $\mathcal{P} (X)$ to be the finitely supported measures on $X$, i.e., $\nu \in \mathcal{P} (X)$ is of the form
$$\nu = \sum_{i=1}^n a_i \delta_{x_i},$...

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### Dehn functions of finitely presented simple groups

Any finitely presented simple group has solvable word problem, and hence recursive Dehn function. I'm curious though how wild these recursive functions could possibly be.
One concrete question is ...

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### Problem 3.14 from Kirby's list

In his famous list of Problems in Low-Dimensional Topology, Kirby states the following as Problem 3.14 (B), which is attributed to Thurston:
Conjecture: Suppose $G$ (an arbitrary group I suppose) ...

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### dichotomy in hyperbolic groups

Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...

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### Property A, Higson-Roe condition and its applications

Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition:
Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ ...

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### Can distinct meridians commute in a knot group?

Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...

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### Examples of Lattices of Sp(n,1)

$Sp(n,1)$ is the isometry group of $n$-dimensional quaternionic hyperbolic space. It is written in literature that the group is an example of a hyperbolic groups. Can you suggest me any reference ...

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### Quotient of an Artin group is an Artin group

I'm working on a problem about Artin groups, and to simplify this problem I want to take a quotient that allow us to go to an easier Artin group, but I'm not sure if the quotient is well defined. This ...

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### Computation of $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$

As a term of a Serre spectral sequence, I would like to compute the cohomology group with compact support $H^*_c(\mathcal{M}_{1,[2]},\mathbb{V}_1)$ of the moduli space of genus $1$ curves with $2$ ...

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### Subgroup membership problem in simple groups

Let $G$ be a finitely presented simple group. By Kuznetsov (1958), $G$ has decidable word problem. However, by Scott [1], $G$ may have undecidable conjugacy problem. Is anything known about other ...

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### Passing to normal forms in graphs of groups

Given a word $w \in X^{\pm 1}$ representing an element of the free group $F(X)$ there is a (usually non-unique) sequence $w=w_0 \to w_1 \to \cdots \to w_r$ with $|w_i|>|w_{i+1}|$ where $w_r$ is the ...

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### Hyperbolic groups and residual finiteness

The existence of a hyperbolic group which is not residually finite is (to my knowledge) an open question. Is there any reason to suspect that all hyperbolic groups are residually finite, perhaps some ...

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### Maps of surfaces to CAT(0) cube complexes

Let $C$ be a locally CAT(0) cube complex. Let $f\colon \Sigma_g \rightarrow C$ be a continuous map from a closed oriented genus $g \geq 1$ surface to $C$. Is it always possible to find a cube ...

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### Writing an element of a free product of $C_2$'s as a product of order-$2$ elements

My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...

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### Does every infinite, connected, locally finite, vertex-transitive graph have a leafless spanning tree?

My question is
Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must
$G$ have the following substructures?
i) a leafless spanning
tree;
ii) a spanning forest consisting ...

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### Groups that don't contain quasi-hyperbolic plane

Is there any known example of a one-ended finitely presented group with exponential growth that does not contain a quasi-isometric copy of the hyperbolic plane?
This question is motivated by the ...

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### Are the canonical embeddings into $G*_AH$ quasi-isometric?

Suppose $A,G,H$ are finitely generated groups and $A$ is quasi-isometrically embedded into $G$ and $H$. Does it follow that the natural embeddings of $G$ and $H$ into $G*_AH$ are quasi-isometric?
I ...

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### Finite homology of a homogeneous space

Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be ...

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### Right-angled Artin groups that split as direct products

For a finite graph $X$, let $A_X$ denote the associated right-angled Artin group. Thus $A_X$ is generated by the vertices of $X$ subject to the relations $[v,w]=1$ whenever vertices $v$ and $w$ are ...

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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 $(X,d)$. ...