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.

Filter by
Sorted by
Tagged with
1
vote
1answer
20 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. ...
2
votes
1answer
92 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 ...
1
vote
1answer
124 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
1answer
43 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 $29$, ...
0
votes
1answer
62 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 ...
1
vote
1answer
42 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 ...
4
votes
1answer
72 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
1answer
193 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 $...
1
vote
2answers
66 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 \...
3
votes
1answer
116 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
1answer
70 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 ...
3
votes
2answers
97 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
1answer
166 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 $\...
2
votes
1answer
100 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 ...
3
votes
1answer
48 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
vote
0answers
51 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 $(x|z)\ge \max\{(x|w),(y|z)\}$...
4
votes
2answers
111 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 ...
2
votes
0answers
111 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 ...
1
vote
0answers
31 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 ...
1
vote
1answer
259 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_{...
1
vote
1answer
22 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 ...
2
votes
2answers
86 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 ...
6
votes
1answer
384 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 ...
7
votes
0answers
214 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 ...
2
votes
1answer
65 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 $g=h^n$.
1
vote
0answers
70 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 ...
3
votes
1answer
128 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 ...
5
votes
1answer
70 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 $T_1$...
0
votes
1answer
86 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 ...
2
votes
1answer
62 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 \...
1
vote
0answers
159 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 ...
2
votes
1answer
266 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 ...
0
votes
1answer
120 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/ ...
2
votes
0answers
181 views

Hyperbolic groups from Dehn functions

Hyperbolic groups may be defined as finitely generated groups admitting a linear Dehn function. I wonder whether it is possible to prove most of the classifical properties of hyperbolic groups in this ...
2
votes
1answer
90 views

Modify a Dehn presentation

Suppose you have a Dehn presentation $\langle X \mid R \rangle$ of (say not the free group) a hyperbolic group. Has there been some work done on changing this presentation, e.g. adding a relation ("...
1
vote
1answer
119 views

Equalities and inequalities for quadrilaterals in hyperbolic space

In euclidean space any quadrilateral satisfies equalities and inequalities $$a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2$$ $$a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2$$ where $a,b,c,d$ are the side lenghts, $...
1
vote
1answer
155 views

Weakly relatively hyperbolic groups

A finitely-generated group $G$ is weakly hyperbolic relatively to a collection of subgroups $\{ H_1, \ldots, H_r\}$ if the graph obtained from a Cayley graph of $G$ by coning off the cosets of the $...
9
votes
1answer
341 views

Totally geodesic hypersurface in compact hyperbolic manifold

In [Zeghib: Laminations et hypersurfaces géodésiques des variétés hyperboliques, Annales scientifiques de l'ENS, 1991] it is shown, that in a compact manifold of negative curvature, there exists only ...
2
votes
1answer
111 views

From Tilings To Groups

I am studying (on my own) some random group theory, and using this primer. The book focuses on finitely presented groups, and the main definition of a hyperbolic group there is "word-hyperbolic", ...
3
votes
1answer
162 views

Gromov's boundary at infinity, drop the hypothesis on hyperbolicity

It's an easy result that if we have two quasi isometric hyperbolic spaces, then their Gromov boundaries at infinity are homeomorphic. I found online these notes where at page 8, prop 2.20 they seem ...
10
votes
3answers
1k views

Are all $\delta$-hyperbolic groups CAT(0)?

In Alessandro Sisto's notes on geometric group theory he mentions that "Many, probably most people in the field" believe that not all $\delta$-hyperbolic groups are CAT(0) groups. Can anything be said ...
1
vote
1answer
153 views

Is $\mathbb{Z}_3$ CAT(0) and/or (Gromov) $\delta$-hyperbolic?

This example is confusing me. Is $\mathbb{Z}_3 = \langle a\vert a^3\rangle$ $\operatorname{CAT}(0)$ and/or (Gromov) $\delta$-hyperbolic? The Cayley graph clearly has bounded diameter, therfore it is ...
2
votes
1answer
73 views

Ultralimit of Cayley graph of $\mathbb{Z}^2$

I am new to ultralimits and I am trying to find out what the asymptotic cone $\operatorname{Cone}_{\omega}(X)$ of $X:=\operatorname{Cay}(\langle\mathbb{Z}^2\vert(1,0),(0,1)\rangle)$ is. And how to ...
2
votes
1answer
153 views

Strong contraction of hyperbolic space

I'm trying to study Hyperbolic geometry, but I can not understand the following statement. Let $X$ be a $δ$-hyperbolic space. Then, there exists $M > 0$ such that for any geodesic $γ$, and any ...
2
votes
1answer
428 views

Thin triangles vs Slim triangles in hyperbolic spaces

What is the difference between thin triangles and slim triangles in $\delta$ hyperbolic spaces? Google search seems to consider thin and slim as synonyms and shows the same results for the two.
2
votes
2answers
436 views

Fundamental group of a closed hyperbolic surface is Gromov hyperbolic

Does anyone have a reference for the proof of the result in the title? Thanks!
2
votes
0answers
47 views

Deck transformations and Gromov Hyperbolicity

I would like to ask, once more, for some references in Gromov-hyperbolic spaces. The question is specifically the following: Does someone know any alternative reference, alternative proof, anything, ...
2
votes
0answers
108 views

Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that $...
2
votes
1answer
334 views

Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
3
votes
2answers
204 views

Is a $0$-hyperbolic group free?

In his article, Abderezak Ould Houcine asks the following question: If $G$ is a hyperbolic group, let $\delta_0(G)$ denote the infinimum of $\delta$ for which $G$ is $\delta$-hyperbolic. When $\...