# 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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### 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. ...
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### $\delta$ 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 ...
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### 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 ...
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### 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 $29$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 \$\...