All Questions
Tagged with gt.geometric-topology geometric-group-theory
220 questions
7
votes
1
answer
213
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
3
votes
1
answer
269
views
$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
- 111
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
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 \...
3
votes
1
answer
100
views
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Let $G_{1}$ and $G_{2}$ be connected semisimple real Lie groups with no compact factors and finite center and let $K_{1}$ and $K_{2}$ denote some fixed choice of their maximal compact subgroups, ...
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
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
8
votes
4
answers
601
views
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-...
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
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
8
votes
2
answers
489
views
Amalgamated product acting on CAT(0) cube complex
I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...
- 349
2
votes
0
answers
135
views
Need help understanding the geometry of a particular building structure
$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...
- 21
3
votes
0
answers
99
views
Relation of geometric and polyhedral convergence
By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
- 61
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
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
9
votes
1
answer
467
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 ...
- 301
2
votes
0
answers
162
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 $\pi(K) = \pi_1(S^3 \setminus K)$...
- 2,533
4
votes
0
answers
188
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 ...
- 301
3
votes
1
answer
193
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 ...
- 31
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 ...
6
votes
1
answer
166
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 $(X,d)$. ...
- 312
2
votes
1
answer
66
views
Sets with a good lift under a covering
Suppose I have a covering map $\pi : E \to X$ between (nice) topological spaces, and $x \in X$. If $U \ni x$ is a very small open set, then $\pi^{-1}(U)$ is a discrete union of subsets $V_d \subset X$ ...
- 6,652
1
vote
1
answer
182
views
A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
- 331
7
votes
1
answer
240
views
If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?
I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 ...
- 1,097
6
votes
0
answers
355
views
Arithmetic Teichmüller curves, first eigenvalue of the Laplacian, McMullen's expander conjecture
$\DeclareMathOperator\SL{SL}$By a result due to Ellenberg and McReynolds, any finite index subgroup $\Gamma$ of $\Gamma(2) \subset \SL\left(2,\mathbb{Z}\right)$ is the Veech group of an arithmetic ...
- 31
2
votes
0
answers
153
views
"Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\CC{C}$In the article "Hyperbolic rigidity of higher rank lattices", Thomas Haettel has proved the following theorem: Let $\Gamma$ be a ...
- 331
3
votes
2
answers
197
views
HNN decomposition of finite rank free group over infinite rank subgroups
It's a nice result of Swarup that whenever a free group $G$ splits as an HNN extension $G = J \ast_{H,t}$ with $H$ a finitely generated subgroup, there exist splittings $J = J_1 \ast J_2$ and $H = H_1 ...
- 302
1
vote
0
answers
98
views
Question about coarse fixed point property in large-scale geometry
I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.
I start with some main definitions from this article. A coarse ...
- 103
13
votes
2
answers
697
views
Mapping Out(F_n) to the mapping class group
Let $\mathrm{Out}(F_g)$ denote the automorphism group of a free group, and $\mathrm{Mod}_g$ the mapping class group of a closed oriented genus $g$ surface. Is there a map, as indicated with the dashed ...
- 40.2k
8
votes
0
answers
432
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
- 111
0
votes
0
answers
111
views
Inverse limit in category of graphs
Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....
- 1
11
votes
1
answer
517
views
Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?
Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-...
- 327
7
votes
2
answers
617
views
Status of the Hopf-Thurston sign conjecture in dimension 4
A famous conjecture in topology asserts:
The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.
This was conjectured by Hopf for manifolds with non-...
- 11.9k
1
vote
1
answer
210
views
Liouville property of hyperbolic spaces
It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am ...
- 1,407
8
votes
1
answer
200
views
For which planar topological spaces $Z$ does there exist a hyperbolic group $\Gamma$ with $\partial \Gamma \cong Z$?
Recall a topological space is called planar if it can be embedded in $S^2$. I'm interested in understanding hyperbolic groups with planar boundaries.
In [1], it is shown that if a one-ended hyperbolic ...
- 629
3
votes
2
answers
344
views
Hyperbolic volume of hyperbolic knots
Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...
- 223
7
votes
1
answer
372
views
Two details from Stallings's proof of the sphere theorem
EDIT: After a little prompting by Mark Grant, I answered the first question in the comments. The second question remains open.
Let $M$ be a compact $3$-manifold with $\pi_2(M) \neq 0$. The sphere ...
- 353
5
votes
1
answer
252
views
Quasi-isometric rigidity of surface groups and commensurability
Let $G$ be a group quasi-isometric to the fundamental group of a genus 2 surface group $H$. It is well known that $G$ is quasi-isometrically rigid, i.e. $G$ and $H$ are virtually isomorphic. Does ...
- 855
6
votes
1
answer
375
views
Non-compact Dirichlet fundamental domains and free Fuchsian groups
Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model.
Assume throughout that $\mathcal{F}$...
- 63
1
vote
0
answers
141
views
What are the order 5 symmetries of the the torus knot T(3,4)?
Generally, I am interested in understanding the free actions of finite cyclic groups on $S^3$ which leave invariant an oriented torus knot $T(p,q)$. For a specific example, consider the knot $T(3,4)$ ...
2
votes
0
answers
139
views
Image of the pure braid group under the Artin presentation into the automorphism group of the nilpotent quotient of a free group?
As I know, it is unknown that the image of the mapping class group of the surface and its Johnson filtration under the higher Johnson homomorphisms.
There are a relationship between the mapping class ...
- 347
4
votes
1
answer
236
views
Mapping class groups of Haken Seifert 3-manifolds (not small)
This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it.
I want to ...
- 309
4
votes
1
answer
249
views
Realizing groups as the fundamental group of graphs of groups allowing non-injective maps?
In the definition of a graph of groups it is assumed that the maps from the edge groups to the vertex groups of injections, however in what follows I will also be interested in the case where the maps ...
- 5,349
8
votes
1
answer
344
views
Characterizations of metric trees
Let $X$ be a geodesic space. Then the following conditions are equivalent:
For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$.
No ...
- 28k