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
901
questions
0
votes
0
answers
49
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 ...
2
votes
0
answers
119
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(...
6
votes
1
answer
191
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 ...
1
vote
0
answers
63
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 ...
1
vote
0
answers
63
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 ...
-1
votes
0
answers
103
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-...
2
votes
1
answer
151
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:...
1
vote
0
answers
75
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 ...
8
votes
2
answers
537
views
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–...
2
votes
0
answers
65
views
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, ...
3
votes
1
answer
261
views
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....
5
votes
1
answer
284
views
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 ...
2
votes
1
answer
93
views
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....
1
vote
2
answers
130
views
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 ...
8
votes
1
answer
358
views
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? (...
16
votes
1
answer
685
views
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 ...
9
votes
1
answer
354
views
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 ...
2
votes
0
answers
111
views
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. ...
0
votes
0
answers
63
views
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 ...
2
votes
1
answer
54
views
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 ...
12
votes
0
answers
177
views
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 ...
5
votes
0
answers
69
views
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 ...
2
votes
0
answers
144
views
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 $g\in G$ ...
5
votes
0
answers
122
views
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 ...
4
votes
1
answer
82
views
$K(\pi,1)$-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 ...
6
votes
1
answer
304
views
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 ...
5
votes
1
answer
200
views
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 ...
5
votes
1
answer
206
views
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 ...
3
votes
0
answers
156
views
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_{\...
12
votes
6
answers
3k
views
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 ...
5
votes
0
answers
241
views
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},$...
17
votes
2
answers
545
views
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 ...
8
votes
1
answer
682
views
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) ...
8
votes
1
answer
384
views
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 ...
2
votes
0
answers
78
views
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 $A$ ...
2
votes
0
answers
124
views
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 $\pi(K) = \pi_1(S^3 \setminus K)$...
4
votes
0
answers
120
views
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 ...
2
votes
1
answer
193
views
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 ...
2
votes
1
answer
308
views
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 $2$ ...
8
votes
2
answers
440
views
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 ...
2
votes
1
answer
191
views
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 ...
5
votes
0
answers
276
views
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 ...
3
votes
1
answer
182
views
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 ...
12
votes
0
answers
438
views
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 ...
16
votes
0
answers
295
views
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 ...
4
votes
1
answer
146
views
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 ...
6
votes
2
answers
171
views
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 ...
3
votes
0
answers
103
views
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 ...
11
votes
3
answers
334
views
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 ...
6
votes
1
answer
122
views
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 $(X,d)$. ...