All Questions
Tagged with geometric-group-theory hyperbolic-geometry
21 questions with no upvoted or accepted answers
13
votes
0
answers
223
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 ...
- 1,574
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
6
votes
0
answers
196
views
A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers
I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that
a) the graph $\Gamma$ is fine,
b) $\Gamma$ is not a tree,
c) not all ...
- 101
6
votes
0
answers
160
views
Maximum relator and hyperbolicity
It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows:
Suppose ...
- 721
6
votes
0
answers
383
views
When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?
Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...
- 2,964
5
votes
0
answers
155
views
Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?
It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
- 481
5
votes
0
answers
183
views
Finitely generated nilpotent groups as cusp groups
I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
- 2,090
5
votes
0
answers
689
views
Visual boundary vs. ideal boundary of hyperbolic manifolds?
I apologize in advance if this is elementary; I have very little experience with hyperbolic manifolds, but I'm using hyperbolic manifolds for part of a current project.
Given a discrete torsion-free ...
- 6,025
4
votes
0
answers
433
views
Convex core and geometric finiteness of negatively curved manifolds
I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...
- 41
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
414
views
Geometric intersection number for product of elements of the fundamental group
Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...
- 1,713
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
2
votes
0
answers
87
views
Hausdorff dimension of radial limit sets for divergence type subgroups
Let $X$ be a proper $CAT(-1)$ space.
Let $\Gamma<Isom(X)$ be a subgroup of divergence type.
Is it true that the Hausdorff dimension of the radial limit set of $\Gamma$ in $\partial X$ is equal to ...
- 2,964
2
votes
0
answers
135
views
Extending continuous functions from $\partial X$ to $X\cup \partial X$
Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial X\to\...
- 329
2
votes
0
answers
212
views
Exotic actions of hyperbolic groups
Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that:
The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$.
For every quasiconvex subgroup $H \leq G$...
- 11.3k
2
votes
0
answers
118
views
Local curvature in a Cayley complex
I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local ...
- 565
1
vote
0
answers
90
views
Weil-Petersson metric with respect to covering
Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\...
- 1,713
1
vote
0
answers
153
views
Topological entropy and pseudo-Anosov dilatation for punctured surface
Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ punctures. Assume $2-2g-n<0$. Let $f$ be a pseudo-Anosov mapping class with dilatation $\lambda_f$. In the introduction (1st page) of the ...
- 1,713
1
vote
0
answers
104
views
Algorithm to generate hyperbolic metric on a compact surface
Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
- 1,713
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
0
votes
0
answers
186
views
Hyperbolic manifold of dim 3 with finite volume.
The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) ...
- 277