# 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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### $\delta$-hyperbolic group is finitely presented

The following corollary is from Discrete groups by Ohshika. Corollary 2.70. Hyperbolic groups are finitely presented. The author didn't prove it but said that 'Combining this theorem with the ...
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### Distance between points on Gromov boundary

$\textbf{Context:}$ Let $K$ be a connected graph endowed with a distance $d$ given by the length of the shortest path between two points. A path $\alpha$ between two vertices $x$ and $y$ is called a ...
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### 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 $\frac{az+b}{cz+d}$ ...
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### 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 ...
1 vote
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### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...