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.

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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 $G$...
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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 $\...
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66 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 ...
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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 ...
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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 $(x|z)\ge \max\{(x|w),(y|z)\}$...
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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 ...
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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 ...
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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 ...
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185 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_{...
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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 ...
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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 ...
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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 ...
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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 ...
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64 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 $g=h^n$.
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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 ...
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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 ...
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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 $T_1$...
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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 ...
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56 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 \...
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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 ...
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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 ...
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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/ ...
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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 ...
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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 ("...
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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, $...
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135 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 $...
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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 ...
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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", ...
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154 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 ...
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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 ...
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133 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 ...
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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 ...
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130 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 ...
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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.
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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!
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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, ...
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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 $...
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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 ...
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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 $\...
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Hyperbolic metric spaces

I'm trying to prove a simple proposition wich is in Burago's "A Course in Metric Spaces" (Exercise $8.4.5$, p.$287$). Before exposing my problem, let me give some definitions. A metric space $(X,d)$ ...
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239 views

Using the Gromov product in inappropriate ways

The Gromov product $(x,y)_z=1/2(d(z,x)+d(z,y)-d(y,x)$ is used in Gromov hyperbolic groups to measure how long two rays stay together or how thin a triangle is. In particular, if $(x,y)_z=n$ in a $\...
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References for Hyperbolic Graph Theory

I'm sorry to disturb you but I really got stuck! I can't find any clear and, somewhat, complete reference for this topic. I'm looking for a book, or review, or survey or course notes regarding "...
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Infinite geodesic rays leaving a K-quasiconvex subgroup stay K-close to it.

I am going through some basic properties of $\delta$-hyperbolic spaces and groups and I am having some difficulties proving precisey some things that are anyway intuitively clear to me. Let $G$ be a $...
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normalizer of a cyclic subgroup in a torsion-free hyperbolic group

How one can show that in a torsion-free hyperbolic group if elements $x$ and $y$ (edit: $y\ne1$) satisfy: $$ xy^mx^{-1}=y^n $$ then $m=n$ and $x$ and $y$ belong to the same cyclic subgroup?
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Parabolic isometries on Gromov hyperbolic spaces

Let $X$ be a $\delta$-hyperbolic geodesic space. Then we have the following classification of isometries on $X$: Theorem: Let $g$ be an isometry on $X$. Then, exactely one of the following case ...
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562 views

Boundary of hyperbolic spaces and isometries

Do you know a good reference about boundary of hyperbolic spaces (following Gromov) and the classification of the isometries acting on hyperbolic space (hyperbolic, parabolic and elliptic isometries)? ...
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Are hyperbolic triangle groups hyperbolic?

This might be a silly question, but are hyperbolic triangle groups hyperbolic, in the sense of Gromov? By a hyperbolic triangle group, I mean a group given by a presentation, $$\langle a, b, c; a^p, ...
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Isoperimetric inequalities of a group

How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being $\delta-$...