All Questions
Tagged with geometric-group-theory hyperbolic-geometry
79 questions
4
votes
1
answer
392
views
Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space
Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂𝑋 be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich https://...
- 313
4
votes
1
answer
390
views
When does a subgroup of $\operatorname{GL}(n, \mathbb Q)$ have a bounded fundamental domain on $\mathbb R^n$?
$\DeclareMathOperator\GL{GL}$Let $G \subset M_{n\times n~}(\mathbb Z)$ be a finitely generated subgroup of $\GL(n,\mathbb Q)$ (i.e. $g\in G$ is an invertible matrix with entries in $\mathbb Z$). Then $...
- 3,472
4
votes
1
answer
256
views
Action of the isometry group of the hyperbolic 5-space
We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+_{5,1}(\mathbb{R})/SO_5(\mathbb{R})=SL_2(\mathbb{H})/Sp^*_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the ...
- 1,692
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
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
3
votes
1
answer
406
views
δ-hyperbolic space
It is known that A δ-hyperbolic space is a geodesic metric space in which every geodesic triangle is δ-thin. (http://en.wikipedia.org/wiki/%CE%94-hyperbolic_space)
The question is that if we remove ...
- 33
3
votes
1
answer
227
views
Density of ends of long words in a hyperbolic group
Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...
- 305
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
1
answer
244
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
2
votes
1
answer
198
views
Equivalence of harmonic measures on hyperbolic groups
Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, ...
- 2,090
2
votes
1
answer
214
views
Subsets of the boundary of a surface group
Consider the surface group $\Gamma=\langle a,b,c,d\mid [a,b][c,d]=1\rangle$: it is a Gromov hyperbolic group; its Gromov boundary $\partial\Gamma$ is homeomorphic to $S^1$ (the unit circle).
I would ...
- 329
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
2
answers
158
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 ...
- 331
1
vote
1
answer
162
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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
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
1
vote
2
answers
310
views
Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?
Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?
- 2,964
1
vote
1
answer
79
views
Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $H$. Does $H$ permute the components of $\partial G - \Lambda H?$
Let $H$ be a co-dimension 1 quasiconvex subgroup of a one-ended hyperbolic group $G$. In particular, $\partial G$ is connected but $\partial G - \Lambda H$ is disconnected. The number of components of ...
- 639
1
vote
1
answer
72
views
orthogonal transformations of one sheeted hyperboloid $S^{1,1}$
I have asked this question few days ago in MathStackExchange but I got only one response which gave a partial answer to my question, so I decided to ask it here.
I am reading Kulkarni's "Proper ...
- 195
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
1
answer
291
views
A question on Cayley graphs and hyperbolic 3-manifolds
There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...
- 841
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