All Questions
Tagged with gr.group-theory geometric-group-theory
671
questions
-1
votes
0
answers
110
views
What does this notation mean? [closed]
For context, $\phi_{f} $ and $\psi_{u} $ are two different automorphisms of the same group. I would like to know what the following notation, $\phi_{f}^{\psi_{u} } $ , is referring to? What does it ...
- 1
3
votes
0
answers
195
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 ...
- 49
1
vote
1
answer
96
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 ...
- 143
0
votes
0
answers
74
views
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 ...
- 21
7
votes
0
answers
83
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 ...
- 3,201
14
votes
1
answer
939
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 ...
- 1,055
2
votes
0
answers
90
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 ...
- 141
0
votes
0
answers
143
views
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 $A_p$ ...
- 121
1
vote
1
answer
193
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 ...
- 21
4
votes
1
answer
114
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 ...
- 447
4
votes
4
answers
248
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 $G$ ...
- 329
7
votes
1
answer
388
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....
- 329
5
votes
0
answers
186
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 ...
- 133
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 ...
- 295
1
vote
0
answers
109
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 $$ G=B S(1, 2)=\left\langle a, t \mid t a
t^{...
- 763
3
votes
0
answers
284
views
Is G(4,7) a Coxeter group
Let $G(4, 7)$ be an abstract group with the presentation
$$\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 ...
- 1,916
2
votes
0
answers
54
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 ...
- 763
2
votes
0
answers
138
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}$, $...
- 763
1
vote
0
answers
150
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 ...
- 273
6
votes
1
answer
223
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 ...
- 63
1
vote
0
answers
69
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 ...
- 3,201
1
vote
0
answers
71
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 ...
- 11
0
votes
0
answers
119
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-...
- 103
2
votes
1
answer
166
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:...
- 309
1
vote
0
answers
124
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 ...
- 41
8
votes
2
answers
582
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–...
- 309
2
votes
0
answers
108
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, ...
- 309
3
votes
1
answer
308
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....
- 309
5
votes
1
answer
290
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 ...
- 309
2
votes
1
answer
98
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....
8
votes
1
answer
388
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? (...
- 283
16
votes
1
answer
726
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 ...
- 763
9
votes
1
answer
361
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 ...
- 93
2
votes
0
answers
118
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. ...
- 177
0
votes
0
answers
64
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 ...
- 61
2
votes
1
answer
55
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 ...
- 537
13
votes
0
answers
192
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 ...
- 1,354
2
votes
0
answers
152
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$ ...
- 185
5
votes
0
answers
135
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 ...
- 51
6
votes
1
answer
375
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 ...
- 63
5
votes
1
answer
214
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 ...
- 125
5
votes
1
answer
212
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 ...
- 245
3
votes
0
answers
162
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_{\...
- 447
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 ...
- 161
17
votes
2
answers
565
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 ...
- 3,201
2
votes
1
answer
202
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 ...
- 447
8
votes
2
answers
443
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 ...
3
votes
1
answer
203
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 ...
- 963
5
votes
0
answers
311
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 ...
- 283
12
votes
0
answers
449
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 ...
- 1,308