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
0 votes
0 answers
4 views

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 ...
user avatar
0 votes
1 answer
58 views

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 ...
user avatar
  • 959
4 votes
0 answers
37 views

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 ...
user avatar
  • 41
0 votes
0 answers
27 views

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. ...
user avatar
2 votes
0 answers
37 views

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 ...
user avatar
2 votes
1 answer
54 views

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 ...
user avatar
1 vote
1 answer
17 views

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 ...
user avatar
  • 859
1 vote
1 answer
60 views

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 ...
user avatar
  • 27
1 vote
0 answers
44 views

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 ...
user avatar
  • 27
1 vote
1 answer
54 views

Bounds of the Gromov product associated with quasi-geodesics

Let $l_1$ be a quasi-geodesic with endpoints $x_0,x_1$ and $l_2$ be a quasi-geodesic with endpoints $x_1,x_2$ in a $\delta$-hyperbolic space. Let $y_i\in l_i$ be a point ($i=1,2$). Suppose $d(y_1,x_0)...
user avatar
  • 311
0 votes
0 answers
22 views

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 $\frac{az+b}{cz+d}$ ...
user avatar
0 votes
1 answer
157 views

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 ...
user avatar
  • 169
1 vote
1 answer
34 views

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 ...
user avatar
  • 163
1 vote
2 answers
108 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. ...
user avatar
3 votes
1 answer
179 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 ...
user avatar
2 votes
1 answer
166 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 ...
user avatar
1 vote
1 answer
61 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 $29$, ...
user avatar
0 votes
1 answer
110 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 ...
user avatar
  • 427
1 vote
1 answer
68 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 ...
user avatar
5 votes
1 answer
202 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 ...
user avatar
12 votes
1 answer
288 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 $...
user avatar
  • 14.8k
1 vote
2 answers
135 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 \...
user avatar
3 votes
1 answer
190 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 ...
user avatar
1 vote
1 answer
172 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 ...
user avatar
3 votes
2 answers
116 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 $G$...
user avatar
1 vote
1 answer
409 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 $\...
user avatar
2 votes
1 answer
151 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 ...
user avatar
3 votes
1 answer
54 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 ...
user avatar
1 vote
0 answers
57 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 $(x|z)\ge \max\{(x|w),(y|z)\}$...
user avatar
  • 1,444
4 votes
2 answers
130 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 ...
user avatar
3 votes
0 answers
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 ...
user avatar
1 vote
0 answers
42 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 ...
user avatar
1 vote
1 answer
436 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_{...
user avatar
  • 3,889
1 vote
1 answer
23 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 ...
user avatar
  • 727
2 votes
2 answers
112 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 ...
user avatar
  • 727
7 votes
1 answer
687 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 ...
user avatar
8 votes
0 answers
251 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 ...
user avatar
  • 825
2 votes
1 answer
67 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 $g=h^n$.
user avatar
2 votes
0 answers
131 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 ...
user avatar
  • 61
3 votes
1 answer
155 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 ...
user avatar
  • 5,896
5 votes
1 answer
96 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 $T_1$...
user avatar
  • 5,896
0 votes
1 answer
125 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 ...
user avatar
  • 5,896
2 votes
1 answer
90 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 \...
user avatar
  • 5,896
1 vote
0 answers
181 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 ...
user avatar
  • 2,950
2 votes
1 answer
350 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 ...
user avatar
0 votes
1 answer
197 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/ ...
user avatar
  • 187
2 votes
0 answers
275 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 ...
user avatar
  • 31.5k
2 votes
1 answer
131 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 ("...
user avatar
  • 1,863
1 vote
1 answer
175 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, $...
user avatar
  • 63
1 vote
1 answer
205 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 $...
user avatar
  • 31.5k