Questions tagged [gromov-hyperbolic-spaces]
Gromov hyperbolic spaces, also known as $\delta$-hyperbolic spaces, are geodesic spaces in which every triangle is thin. Hyperbolic groups are fundamental examples of Gromov hyperbolic spaces in geometric group theory.
76
questions
0
votes
0
answers
19
views
Definition of convergence for the Gromov boundary
BirdsongHaeflinger "Metric spaces of non-positive curvature" defines the Topology of the compactification $\bar X=X\cup \partial X$ via generalized rays. A sequence $c_n$ of such rays ...
- 26
0
votes
0
answers
13
views
Bi-infinite geodesic in geodesic Gromov-Hyperbolic spaces
For a geodesic Gromov Hyperbolic metric space X is it true that there exists $C>0$ such that any two bi-infinite geodesic with same end points at boundary stays within $C$-neighbouhood of each ...
- 1,978
1
vote
0
answers
59
views
A question about Quasi Hyperbolic metric
I am reading Quasicoformally Homogeneous Domains by F. W. Gehring t And B. P. Palks
Let $D$ be a proper subdomain of $\mathbb{R}^n.$ Define the function $\rho.(x) = \frac{1}{dist(x, \partial D)}.$ ...
- 11
2
votes
0
answers
22
views
Negatively curved pair of pants approximation of a tripod
A pair of pants is a surface homeorphic to the $3$-punctured sphere. It is a well known fact that a pair of pants admits a metric of constant curvature $-1$. Usually one is interested in pairs of ...
- 951
0
votes
0
answers
11
views
Induced homeomorphism from a quasi-isometry between hyperbolic spaces
Theorem.
Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$.
If $X$ and $Y$ are proper, then $\phi$ induces a homeomorphism between their boundaries.
The ...
- 101
0
votes
1
answer
88
views
Tesselation of hyperbolic plane are hyperbolic
Let’s take a tesselation $T$ of the hyperbolic plane (not necessarily regular), my intuition tells me that clearly $T$ should be hyperbolic itself (in the sense of Gromov or using $\delta$-slim ...
- 1,009
4
votes
0
answers
58
views
Real tree and simplicial tree
A real tree is a metric space $(X,d)$ such that for any points $x,y\in X$ there is a unique path from $x$ to $y$, and which is a geodesic. Equivalently, it is a $0$-hyperbolic space.
A simplicial tree ...
- 41
0
votes
0
answers
30
views
Geodesic traingle is contained within the $3\delta$ ball centered at any of its vertex
Given any vertex of a geodesic triangle in hyperbolic space, show that the entire triangle is contained in the $3\delta$ ball centred at that vertex, where $\delta$ is the constant of hyperbolicity.
...
- 570
2
votes
0
answers
69
views
Is there always a natural correspondence between the ends of a geodesic space $X$, and the connected components of $\partial_\infty X$?
For a metric space $X$, let $\partial_\infty X$ denote the (Gromov) boundary at infinity of $X$. The following fact appears as an exercise in Bridson-Haefliger's standard text [1, p. 430].
Let $X$ be ...
- 674
2
votes
1
answer
80
views
Gromov product on the boundary
I was reading the book Elements of Asymptotic Geometry by Sergei Buyalo and Viktor Schroeder. There in the second chapter Gromov boundary is defined as the equivalence classes of sequences (in the ...
- 1,978
1
vote
1
answer
19
views
How to quasify a proposition that holds for trees to general hyperbolic spaces?
In Géométrie et théorie des groupes : les groupes hyperboliques de Gromov, ch. 8, the authors describe how one can approximate a hyperbolic space that satisfy some finiteness condition with a metric ...
- 951
1
vote
1
answer
70
views
Distance between geodesic rays 2
This is a follow up to my previous question which turned out to be wrong. Now my question instead is this:
Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side ...
- 51
1
vote
0
answers
55
views
Distance between geodesic rays
I am trying to prove the following but so far did not succeed.
Given a $\delta$-hyperbolic space $(X,d)$ (with Rips triangle condition: a point on any side is contained in
the $\delta$-neighbourhood ...
- 51
1
vote
1
answer
64
views
Bounds of the Gromov product associated with quasi-geodesics
Let $l_1$ be a quasi-geodesic with endpoints $x_0,x_1$ and $l_2$ be a quasi-geodesic with endpoints $x_1,x_2$ in a $\delta$-hyperbolic space. Let $y_i\in l_i$ be a point ($i=1,2$). Suppose $d(y_1,x_0)...
- 325
0
votes
0
answers
30
views
A in PSL(2,C) is composed of two inversions
I wanna show that if $A \in PSL(2,\mathbb{C})$ then $A= i_{c_1} \circ i_{c_2}$ with $i_{c_j}$ inversion of the sphere $c_j$.
I tried to show it so, I can to identify $A$ with $\frac{az+b}{cz+d}$ ...
- 11
0
votes
1
answer
219
views
Show that this quasi-geodesic ray is not Gromov hyperbolic?
Consider the spiral (t, log(1+t)) (given in polar coordinates); it inherits the Euclidean metric from the plane. I have to show that this spiral (a quasi-geodesic) is not Gromov hyperbolic. In other ...
- 325
1
vote
1
answer
39
views
Bounded Geodesics in a Quadrangle of Fixed Length.
The following question is a slightly weaker version of the question presented here: Geodesic Quadrangle in a Hyperbolic Space. However, this bound suffices for the statement mentioned in the original ...
- 173
1
vote
2
answers
149
views
Hyperbolic boundaries of infinitely generated groups
I was wondering the following: in every book and paper that I looked into the definition of word hyperbolic groups (in the sense of Gromov) contains the condition that the group is finitely generated. ...
- 869
3
votes
1
answer
217
views
$\delta$ thin trianges implies solvable conjugacy problem for hyperbolic groups: Confusion about $\delta$-rectangles
I am trying to understand the proof that a linear dehn function implies solvable conjugacy. I am referring to Notes on solvable and automatic groups by Michael Batty, after Panagiotis Papasoglu. Here ...
- 8,116
2
votes
1
answer
182
views
Examples of hyperbolic groups that have boundary homeomorphic to $S^2$?
I am working on understanding Cannon's Conjecture which is the following:
Suppose that $G$ is an infinite, finitely presented group whose Cayley graph is Gromov-hyperbolic and whose space at infinity ...
1
vote
1
answer
70
views
Understanding the proof of solvable conjugacy problem for hyperbolic groups.
https://www.math.ucdavis.edu/~kapovich/280-2009/hyplectures_papasoglu.pdf
https://courses.maths.ox.ac.uk/node/view_material/48431
In the first link the theorem I am talking about is on page $29$, ...
- 397
0
votes
1
answer
118
views
Extending a quasi-isometry of a neutered hyperbolic space
Suppose $\phi : B \to B$ is a quasi-isometry of a neutered space $B$ (so $B$ is obtained by removing a collection of disjoint open horoballs from $\mathbb{H}^n$, and the metric $d_B$ on $B$ is the ...
- 427
1
vote
1
answer
90
views
Arzelà-Ascoli for hyperbolic spaces with natural boundary
I am investigating Paulin's method of certain limits of actions on hyperbolic spaces being (in some sense) actions on $\mathbb{R}$-trees.
Let $G$ be a finitely generated group. Part of the proof is ...
- 435
5
votes
1
answer
314
views
Writing an algorithm solving the word-problem in hyperbolic groups
I am reading in the “Metric Spaces of Non-Positive Curvature Book by André Haefliger and Martin Bridson”, on Dehn's Algorithm (Chapter III.Γ, p.449).
Let $\mathcal{A}$ be a finite generating set of ...
12
votes
1
answer
346
views
Are all almost virtually free groups word hyperbolic?
Suppose $G$ is a finitely generated group with a finite symmetric generating set $A$. Lets define Cayley ball $B_A^n := (A \cup \{e\})^n$ as the set of all elements with Cayley length (in respect to $...
- 15.1k
1
vote
2
answers
148
views
What are the word hyperbolic affine Coxeter groups?
It is well-knwon that all affine (irreducible) Coxeter systems can be classified by their Coxeter graphs, see Wikipedia. The corresponding diagrams are $(\tilde{A}_n)_{n \geq 1}$, $(\tilde{B}_n)_{n \...
- 869
3
votes
1
answer
210
views
a long product of elements in hyperbolic group is not a proper power
Let $G$ be a hyperbolic group, i.e., there exist $\delta>0$ and a finite generating set $S$ of $G$ such that the Cayley graph $X$ of $G$ relative to $S$ is a $\delta$-hyperbolic space. Assume also ...
1
vote
1
answer
239
views
Type of an isometry of a $\delta$-hyperbolic space
Let $(X,d)$ be proper geodesic $\delta$-hyperbolic metric space.
Let $\gamma \in Isom(X)$. Denote by $\partial X$ the boundary at infinity of $X$ (which is invariant of base-point).
Let $x\in X$. We ...
4
votes
2
answers
122
views
Proof of property of relatively hyperbolic groups on Wikipedia
The Wikipedia page for "Relatively hyperbolic group" lists this as a property of relatively hyperbolic groups:
"If a group $G$ is relatively hyperbolic with respect to a hyperbolic group $H$, then $G$...
1
vote
1
answer
526
views
All Gromov-hyperbolic spaces are CAT(0)
I want to show that every $\delta-$hyperbolic space is CAT(0), by the definition I'm using $X$ is $\delta-$hyperbolic if the following inequality is satisfied for all $w,x,y,z\in X$ and for some $\...
- 535
2
votes
1
answer
187
views
Torsion element of a non-elementary Hyperbolic group
Let $\Gamma$ be a non elementary hyperbolic group acting on the Gromov boundary $\partial\Gamma$. Let $a \in \Gamma$ be a torsion element i.e $\langle a\rangle$ is finite. Does $a$ fix every element ...
- 12.4k
3
votes
1
answer
60
views
The collapsing map and its coarse inverse are $32 \delta$-coarse Lipschitz to each other
TL,DR: Why do we have $$d(\overline{\kappa} \circ \kappa, Id) \leq 32 \delta?$$
I am reading Chapter 11 from the book "Geometric group theory" by Cornelia Druţu and Michael Kapovich (freely available ...
- 1,286
1
vote
0
answers
59
views
Gromov hyperbolicity of Metric Spaces
Proof of lemma 2.4
Hi, I was doing some self reading on Gromov geometry and I have difficulty accepting the proof given above for lemma 2.4. While I can understand that $(x|z)\ge \max\{(x|w),(y|z)\}$...
- 1,452
4
votes
2
answers
155
views
What surfaces(manifolds) can be the boundary of hyperbolic groups?
Question: What surfaces can be the Gromov boundary of a hyperbolic group? (You could also ask the same question except for higher dimensional manifolds.)
I know that spheres appear as the boundary of ...
user29123
3
votes
0
answers
142
views
Gromov hyperbolic Space example
I'm reading the original paper of Gromov Hyperbolic Groups. There, he gives the next example
Let $X_0,d$ be an arbitrary metric space ande let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ be a positive ...
- 61
1
vote
0
answers
44
views
Wedderburn's little theorem from a superalgebra point of view - reformulation from upper half-plane $H$ instead of $R^2$
If in standard algebra
every finite division ring is a field
from a superalgebra point of view what is the correspondent formulation to say that every "super-finite-division-ring" is a superfield ...
- 165
1
vote
1
answer
521
views
Geodesic ray converges to infinity
I am reading this paper on boundaries of hyperbolic groups. In this paper, a geodesic metric space $(X,d)$ is considered. A sequence of points $(x_n)_{n \geq 1}$ converges to infinity if
$$\lim \inf_{...
- 4,256
1
vote
1
answer
27
views
Are all connected Gromov hyperbolic spaces also geodetically connected
Again as the title suggests is the above true? If not what examples are there. My only background regarding these spaces have to deals with Hadamard manifolds, which is why I am seeing if its true in ...
- 757
2
votes
2
answers
125
views
Are all Gromov hyperbolic spaces proper metric spaces
Here a proper metric space is a metric space such that all closed balls are compact. My question is are all Gromov hyperbolic spaces proper metric spaces? I only know the rudimentary definitions of ...
- 757
7
votes
1
answer
808
views
Topology on the Gromov Boundary of a Hyperbolic Space
Let $X$ be a proper geodesic metric space that is $\delta$-hyperbolic.
Definition. We define the Gromov boundary $\partial X$ of $X$ as the set of all the geodesic rays $c:[0, \infty)\to X$, where ...
- 17.6k
8
votes
0
answers
259
views
Hyperbolic metric spaces 2
I am trying to prove a lemma in Burago's "A Course in Metric Spaces" (Exercise 8.4.4, p.286). Here is a link to a different person's question about the very next exercise in that book, which also ...
- 835
2
votes
1
answer
67
views
Divisible elements in CAT(0) groups
Given a group $G$ acting on a $CAT(0)$ complex $X$ by isometries can $G$ contain a divisible element, i.e. an element $g\in G$ such that $\forall n\in\mathbb N$ there is $h\in G$ such that $g=h^n$.
- 23
2
votes
0
answers
142
views
Geodesic Quadrangle in a Hyperbolic Space
I'm trying to follow a proof of the fact that if $g$ is an element of a hyperbolic group $G$ with infinite order, then $\langle g \rangle$ is an undistorted subgroup of $G$. The proof relies on the ...
- 61
3
votes
1
answer
166
views
A non-positively curved cube complex that admits a local isometric embedding into a Salvetti complex is special.
I am trying to prove the following:
"A non-positively curved cube complex $X$ that admits a local isometric embedding (that maps cubes to cubes) into the Salvetti complex of some right-angled Artin ...
- 6,081
6
votes
1
answer
124
views
Geodesic quadrangles in CAT($0$) spaces
I am trying to show that any geodesic quadrangle $Q$ in any CAT($0$) space $X$ has a comparison quadrangle in $\mathbb{R}^2$ (same definition as for triangles). One can split $Q$ in two triangles $T_1$...
- 6,081
0
votes
1
answer
143
views
If a group acts properly and coboundedly on a hyperbolic space, each finite subgroup has a (uniformly) bounded orbit.
I am trying to solve the following problem:
"Let $G$ be a group acting properly, coboundedly and by isometries on a hyperbolic space $X$. Show that there is a constant $C$ such that any finite ...
- 6,081
2
votes
1
answer
95
views
Barycentre of bounded subset of hyperbolic space
The exercise I am trying to solve is the following:
"Let $X$ be a $\delta$-hyperbolic space and $A \subseteq X$ a bounded subset with diameter $R$. Show that there exists $p \in X$ such that $A \...
- 6,081
1
vote
0
answers
193
views
Proof of $\delta$-Hyperbolicity of $\mathbb H^n$ just with the hyperboloid model?
Do you know any proof of the fact that $\mathbb H^n$ is Rips-hyperbolic (i.e., geodesic triangles are $\delta$-slim for some $\delta$, also called "Gromov-hyperbolic" in some contexts), which makes no ...
- 3,060
2
votes
1
answer
394
views
Is the Cayley graph of a word-hyperbolic group a CAT(0) metric space?
It is mentioned on the Wikipedia article for Hadamard spaces that the Cayley graphs of a word-hyperbolic (f.g.) group are CAT(0) metric spaces. Is it so? My question comes from the fact that the ...
- 111
0
votes
1
answer
223
views
quasi-geodesics in hyperbolic space
I've stumbled across a proof of geodesic stability in hyperbolic space, located in the following blog post:
https://lamington.wordpress.com/2010/05/19/hyperbolic-geometry-notes-5-mostow-rigidity/
...
- 307