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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How to understand the compactification of the upper half plane as motivation for Gromov compactification of an arbitrary $\delta$-hyperbolic space??
Let $X$ be a $\delta$-hyperbolic space. Let us define an equivalence relation on geodesic rays $\gamma_1, \gamma_2 :[0,\infty) \rightarrow X$ by {$\gamma_1 \sim \gamma_2\,\, \iff d(\gamma_1(t),\...
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Definition of convergence for the Gromov boundary
BirdsongHaeflinger "Metric spaces of non-positive curvature" defines the Topology of the compactification $\bar X=X\cup \partial X$ via generalized rays. A sequence $c_n$ of such rays ...
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Bi-infinite geodesic in geodesic Gromov-Hyperbolic spaces
For a geodesic Gromov Hyperbolic metric space X is it true that there exists $C>0$ such that any two bi-infinite geodesic with same end points at boundary stays within $C$-neighbouhood of each ...
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A question about Quasi Hyperbolic metric
I am reading Quasicoformally Homogeneous Domains by F. W. Gehring t And B. P. Palks
Let $D$ be a proper subdomain of $\mathbb{R}^n.$ Define the function $\rho.(x) = \frac{1}{dist(x, \partial D)}.$ ...
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Negatively curved pair of pants approximation of a tripod
A pair of pants is a surface homeorphic to the $3$-punctured sphere. It is a well known fact that a pair of pants admits a metric of constant curvature $-1$. Usually one is interested in pairs of ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)...
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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 ...
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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 ...
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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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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 $...
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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 \...
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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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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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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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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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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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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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