All Questions
Tagged with gr.group-theory gt.geometric-topology
387 questions
3
votes
1
answer
157
views
Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
12
votes
1
answer
323
views
Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
The title is pretty much the whole question. Let $S_g$ be a closed, oriented surface of genus $g$. Does there exist $n$ such that the mapping class group $\mathrm{Mod}(S_g)$ embeds as a subgroup of $\...
- 3,740
16
votes
2
answers
602
views
$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 911
8
votes
1
answer
349
views
Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
24
votes
1
answer
862
views
The congruence subgroup property for mapping class groups and a conjecture of Grothendieck
This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.
Recall that a subgroup $K$ of a ...
- 25k
9
votes
1
answer
401
views
Cohomological gap in arithmetic groups
$\DeclareMathOperator\SL{SL}$For the sake of this question, let's say that a group $G$ of finite cohomological dimension $n$ has a cohomological gap if for some $0 < i < n$ there is no subgroup $...
- 423
7
votes
1
answer
226
views
When is the action of a mapping class group on the set of punctures realized by a finite subgroup of mapping classes?
I have a curious question about a natural sequence, which I haven't seen answered in the literature.
Let $\Sigma$ be an oriented surface of genus $g$ without boundary with a set $\mathcal P_n$ of $n$ ...
- 318
3
votes
1
answer
160
views
Does the fundamental group of a compact 3-manifold induce full profinite topology on the fundamental group of its boundary?
Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
- 605
11
votes
1
answer
330
views
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 ...
- 43.2k
6
votes
1
answer
200
views
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?
- 103
3
votes
1
answer
177
views
Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
- 254
2
votes
0
answers
100
views
Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
- 21
8
votes
1
answer
217
views
Bilinear forms on abelian $p$-groups: Images of weak metabolizers in the Frattini quotient
Suppose $G$ is a finite abelian $p$-group for $p$ an odd prime and $b : G \times G \to \mathbf{Q}/\mathbf{Z}$ is a non-degenerate symmetric bilinear form. A subgroup $H \leq G$ is a weak metabolizer ...
- 365
7
votes
2
answers
353
views
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}(\...
- 605
7
votes
1
answer
445
views
What is known/expected on the co-growth series of the braid group?
The co-growth series of finitely generated group with respect to generating set $S$ is generating function for the number of words of length $n$ which are equal to 1 in the group.
Its studies ...
- 24.9k
5
votes
0
answers
96
views
$M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?
Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...
- 605
11
votes
1
answer
749
views
Does some knot group fail to surject onto any alternating group?
Are there known examples of knot groups that do not surject onto any alternating group? I would be a little surprised if the answer is negative.
Update: To reflect the interesting part of the question,...
- 2,083
6
votes
1
answer
311
views
Loop manipulation subgroup of the braid group
Recently, I came across a subgroup of the braid group $B_{2n}$ that I'm calling the "loop manipulation" group $H_n$.
The idea is that we treat pairs of adjacent strands in the braid group as ...
- 647
2
votes
1
answer
174
views
Order of a loop around a cone point
Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
- 585
0
votes
1
answer
220
views
Fixed points free automorphisms of Teichmüller spaces
Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
9
votes
1
answer
394
views
Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $F_2$ be a free group of rank 2. There is a surjection $\Aut(F_2)\rightarrow \GL(2,...
- 7,527
2
votes
1
answer
232
views
An interior cone condition for Teichmuller spaces
Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
13
votes
4
answers
747
views
Prove these are not surface groups
For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation:
$$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$
For $n = 1$, ...
- 131
1
vote
0
answers
92
views
$L^p$-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 ...
- 4,432
5
votes
1
answer
356
views
Lattice generated by parabolics
Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free.
For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
- 460
0
votes
0
answers
185
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), ...
- 1
3
votes
0
answers
228
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 ...
- 181
4
votes
1
answer
251
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 ...
- 911
7
votes
1
answer
525
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....
- 349
5
votes
1
answer
206
views
Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic space
Let $\Gamma$ be a hyperbolic group. Let $g$, $\gamma\in \Gamma$ freely generate a non-abelian semigroup (in particular, they don't commute and have infinite order). Does the equation $g\gamma^n=h^m$ ...
- 287
5
votes
0
answers
249
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 661
3
votes
0
answers
308
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 ...
- 2,083
7
votes
1
answer
275
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 ...
- 73
0
votes
0
answers
165
views
Weyl groups are Coxeter groups proof
I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups.
Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...
- 139
4
votes
1
answer
301
views
Understanding $(\mathbb{Z}/3)^2 \times_{\mathbb{Z}/3} M$
I'm currently reading "Bordism of Elementary Abelian Groups via Inessential Brown-Peterson Homology" by Hanke (arXiv:1503.04563) and have come across some notation that I'm not familiar with....
- 545
9
votes
1
answer
377
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
5
votes
1
answer
242
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 ...
- 307
8
votes
1
answer
304
views
The growth rate of a commutator set in a non-elementary group
Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|_S\le n\}$ for any $n\in \mathbb ...
- 145
3
votes
1
answer
158
views
Cohomology of cocompact lattices in hyperbolic spaces
I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
- 131
6
votes
1
answer
290
views
Does the Shalen-Wagreich lemma holds for non-symmetric generating sets?
Let $G$ be a group and $H$ a subgroup with finite index $d$. Then for any finite generating set $S$ of $G$, does $S^{\le k}$ contain a generating set of $H$ where $k$ is a constant depending only on $...
- 145
2
votes
1
answer
314
views
How to get a presentation of the mapping class group of the $n$-punctured sphere
$\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p_1,\...
7
votes
2
answers
269
views
Surjections from genus $n$ surface group to free group of rank $n$
Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
- 73
0
votes
0
answers
91
views
Terminology for discrete subgroups of PSL(2,k), where k is a non-archimedean local field
$\DeclareMathOperator\PSL{PSL}$I'm asking about terminology for discrete subgroups of $\PSL(2,k)$, where $k$ is a non-archimedean local field.
As it is rather clumsy to have to use such expressions ...
- 586
3
votes
0
answers
115
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 ...
1
vote
1
answer
277
views
Ways to prove that $n$-component Brunnian link is nontrivial
The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
- 214
10
votes
3
answers
563
views
Centre of orbifold fundamental group of torus (Klein bottle) with one cone point
$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,
$$\...
- 585
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 613
1
vote
0
answers
231
views
Presentation complexes with same homology and different fundamental groups
If we start with a perfect group $G$ of deficiency zero then there is a presentation $P$ of $G$ such that the number of relations and the number of generators for $P$ are the same. For such $P$, the ...
- 179
11
votes
0
answers
220
views
On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
- 42.4k
1
vote
1
answer
249
views
Name for extension of the symplectic group
Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\...
- 965