# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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### 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 ...
1 vote
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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 ... 133 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 ...
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