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Questions tagged [hyperbolic-groups]

A hyperbolic group is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry.

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Is singular disc diagram same as Van Kampen diagram ? How to draw singular disc diagram for hyperbolic groups?

I couldn't find an elaborate description to draw singular disc diagram for hyperbolic groups (or word hyperbolic groups) and some of the google search result is showing that it similar as Van Kampen ...
mrinal nath's user avatar
1 vote
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Short generators of non-elementary subgroups of hyperbolic groups

In a project I am working on I need to find a pair of elements $a,b$ of a one-ended $\delta$-hyperbolic group $G$ such that the subgroup $\langle a,b \rangle$ is not virtually cyclic. Since the group $...
Michal Ferov's user avatar
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Dehn presentation implies finitely many conjugacy classes of elements of finite order

Let $G$ be a finitely generated hyperbolic group. Show that $G$ contains only finitely many conjugacy classes of elements of finite order. In “Geometric Group Theory: An Introduction” by Clara Löh, it ...
cede's user avatar
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Is the fundamental group of a closed orientable hyperbolic $3$-manifold isomorphic to a non-discrete subgroup of $\mathrm{PSL}(2, \mathbb{C})$?

Consider the fundamental group $\pi_1(M)$ of a closed orientable hyperbolic $3$-manifold $M$. Certain identifications $\tilde{M} \approx \mathbb{H}^3 \approx \mathrm{PSL}(2, \mathbb{C}) / \mathrm{Stab}...
Geoffrey Sangston's user avatar
5 votes
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Number of groups with a bounded short presentation

How many groups there are (up to isomorphism) with a presentation with at most $n$ generators and with relators of length at most $3$? I don't expect there exist a sharp solution, since I know that ...
Dinisaur's user avatar
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How is Conway numbering the hyperbolic groups in his Table 18.1? [closed]

Conway lists many hyperbolic groups in his Table 18.1 on pages 239-240 of The Symmetries of Things. Here are scans of those pages: and The groups are sorted in decreasing order of their ...
Lyle Ramshaw's user avatar
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Quasi-geodesic rays are closed to geodesic rays in proper hyperbolic geodesic spaces

We define the boundary of a hyperbolic metric space $\partial X$ as the equivalence classes of geodesic rays up to finite Hausdorff distance and $\partial_q X$ as the equivalence classes of quasi-...
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Gromov product and distance from a vertex to the opposite line of triangle

I am recently reading definitions of Gromov hyperbolicity. I got stuck on a "trivial" question, that is, given a geodesic triangle $\Delta(x,y,z)$ in any metric space $X$ show that $d(x,[y,z]...
quuuuuin's user avatar
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103 views

$\delta$-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 ...
one potato two potato's user avatar
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Hyperbolic groups are Hopfian

I am writing my master thesis about the following article: https://arxiv.org/abs/2002.10278 On page 14, Fujiwara and Sela give a proof that hyperbolic groups are Hopfian. They say that de la Harpe (&...
TheMathematician's user avatar
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Definition of hyperbolic elements and axes in a (limit) group

I am writing on my master thesis at the moment and it is based on the following article of Fujiwara and Sela: https://arxiv.org/abs/2002.10278 On page 7 they use the terms "hyperbolic element&...
TheMathematician's user avatar
2 votes
1 answer
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Every proper and cobounded action is acylindrical.

I am trying to show every proper and cobounded action is acylindrical. Given any $\epsilon> 0$, we need to show there exist $R, N > 0$, s.t. for every two points $x, y$ with $ d(x, y)\ge R$, ...
Kat's user avatar
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Motivation for Hyperbolic Groups - Soft Question

I took a Geometric Group Theory course this semester. A very big part was hyperbolic groups. What I felt was a little bit lacking in this course was - why do I need hyperbolic groups? What is the ...
user2582354's user avatar
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$xy^2x^{-1}=y^3$ implies $[xyx^{-1},y]=1$

I want to show that if $x,y$ are elements of an hyperbolic group $G$, and $xy^2x^{-1}=y^3$, then $[xyx^{-1},y]=1$. I tried to show that by using triangle thinness but I reached nothing. Another ...
Alessandro Cigna's user avatar
4 votes
2 answers
120 views

Is There an Aspherical 3-Manifold with $H_i(M^3) \cong \mathbb{Z}$ for i = 0 to 3? [closed]

The title says it all: Is there an aspherical 3-manifold with $H_i(M^3) \cong \mathbb{Z}$ for i = 0 to 3?
Jeffrey Rolland's user avatar
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Understand definition of group of hyperbolic motion

The group of hyperbolic motions is defined as the subgroup of S($\mathbb{H}^{2}$) (is the symmetric group of $\mathbb{H}^{2}$) which is generated by PGL($\mathbb{R})^{+}$ (matrices with positive ...
Eli's user avatar
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4 votes
1 answer
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Normal subgroup of triangle group in GAP

Consider the hyperbolic (extended) triangle group $\Delta(2,3,7)=\langle a,b,c\mid a^2,b^2,c^2,(ab)^2,(bc)^3,(ca)^7\rangle$. I construct it in GAP as a finitely presented group, using the standard ...
MathPhysGeek's user avatar
6 votes
1 answer
269 views

Growth of balls vs growth of spheres in hyperbolic groups

Let $G$ be a finitely-generated group equipped with a word-metric. Let $B_n$ and $S_n$ be the $n^{\mathrm{th}}$-ball and $n^{\mathrm{th}}$-sphere, respectively, with respect to the given metric. ...
HUO's user avatar
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Non-elementary Fuchsian Group and the Invariant Measure for the Corresponding Action on the Limit Set

Let $\Gamma$ be a Fuchsian Group (that means a Discrete Subgroup of $RSL_2(\mathbb{R})$) acting on the Closed Unit Disc $\mathbb{D}$. Let $\Lambda (\Gamma)$ be the Limit Set of $\Gamma$, i.e., it is ...
Neil hawking's user avatar
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Sur les Groupes Hyperboliques d’après Mikhael Gromov, English Version

I would like to enquire whether or not the Book Sur les Groupes Hyperboliques d’après Mikhael Gromov for the authors Étienne Ghys and da la Harpe (that is written in French) has been translated into ...
Neil hawking's user avatar
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The Fixed Points of an Elliptic Element in a Kleinian Group

Let $\Gamma$ be a Kleinian Group (a discrete subgroup of $PSL_2(\mathbb{C})$) acting on $\mathbb{D}=\{z \in \mathbb{C} : |z| \leq 1\}$. Let $\Lambda(\Gamma)$, $\Omega(\Gamma)$ be the Limit Set and the ...
Neil hawking's user avatar
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Subgroup membership in hyperbolic groups and determining hyperbolicity

I have a set of two (not necessarily entirely related) questions about hyperbolic groups - I'll ask them as one question since they seem related enough and I have a feeling someone who knows the ...
user101010's user avatar
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1 vote
1 answer
99 views

Distance labels in regular hyperbolic tilings

Consider the order-4 pentagonal tiling of the hyperbolic plane (shown in the figure Hyperbolic plane tiling with pentagons). Pick a vertex $s$ (in white), label it with $0$ and then label all the ...
Matteo Pariset's user avatar
4 votes
4 answers
314 views

An easy example of a non-quasiconvex subgroup

Let $G$ be a finitely generated group, and consider the surjection $\mu:F(A)\to G$ induced by the set of generators $A$, where $F(A)$ is the free group on $A$. A word $w$ is said to be ($\mu$-)...
Dinisaur's user avatar
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3 votes
1 answer
133 views

Finite generation of vertex groups of a cyclic splitting of a hyperbolic group and generalisations of Grushko Theorem

Let $G$ be a finitely generated word hyperbolic group. Suppose $G$ acts non-trivially (without a global fixed point) on a tree without inversions and with cyclic edge stabilizers. Is it true that the ...
24601's user avatar
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1 vote
2 answers
225 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. ...
worldreporter's user avatar
1 vote
1 answer
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Presentation for a hyperbolic group with 2-sphere boundary.

I am looking for examples of hyperbolic groups that have boundary homeomorphic to the 2-sphere, $S^2$. I would like an explicit presentation of such a group so that I can draw its Cayley graph and ...
Chrystal Math's user avatar
8 votes
1 answer
464 views

Is the braid group hyperbolic?

The braid groups satisfy a number of properties that one would expect of a hyperbolic group, liking having a solvable word problem, and having exponential growth. Are the braid groups hyperbolic ...
Quizzical's user avatar
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1 answer
241 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 ...
Chrystal Math's user avatar
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334 views

Trace role in classification of SL(2,R) matrices.

I am working on J.Bochi and Avila's article "Uniformly hyperbolic finite-valued $SL(2,\mathbb{R})$-cocycles".As you may know,there is classification of $SL(2,\mathbb{R})$ which depends on the Trace of ...
pershina olad's user avatar
2 votes
1 answer
138 views

What is the name for the group of hyperbolic rotations in n-dimensional Euclidean space?

In the case of $n = 2$, a hyperbolic rotation matrix by an arbitrary angle looks like: $\begin{bmatrix} \cosh(\theta) & \sinh(\theta)\\ \sinh(\theta) & \cosh(\theta) \end{bmatrix}$ $\...
Mr X's user avatar
  • 907
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1 answer
174 views

Groups acting properly discontinuous and cocompactly on the hyperbolic plane by isometries.

I'm looking for a non-elementary hyperbolic group which is quasi isometric to $\mathbb{H}^2$ (and if possible one quasi-isometric to $\mathbb{H}^3$). I know the group $\text{PSL}(\mathbb{R})$ acts by ...
Tychonoff3000's user avatar
2 votes
1 answer
306 views

Examples of Hyperbolic Groups

We already have the Milnor-Svarc Lemma, which tells us that if a group acts "nicely" on a space, then the Cayley graph of the group is quasi-isometric to the given space. This gives us a lot of ...
ZxJx's user avatar
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5 votes
0 answers
298 views

Function on the Cartesian product of group-orbits

Let $\Gamma$ be a group generated by two matrices as follows: $\Gamma:= \bigg\langle \begin{bmatrix}1&0\\3&1\end{bmatrix},\begin{bmatrix}13&12\\12&13\end{bmatrix} \bigg\rangle$ For ...
Student's user avatar
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2 votes
1 answer
237 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 ...
tattwamasi amrutam's user avatar
0 votes
1 answer
182 views

Group generated by all inversions in hyperbolic lines

In Groups and Geometry by Lyndon, Chapter 9 part 3 (page 165) started by introducing a new group (denoted as $\widetilde{H}$ below) and a theorem. Let $H$ denote the hyperbolic group and $H^{+} = \{ ...
user avatar
1 vote
1 answer
101 views

G-invariance of set of points fixed by loxodromic elements in G

Let G be a non-elementary subgroup of Mobius transformations. How can we show that the set of points fixed by loxodromic elements in G is G-invariant? I proved it by directly computations, but I ...
Student's user avatar
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0 votes
1 answer
659 views

About limit set of non-elementary Fuchsian group

I am reading a note on hyperbolic surfaces (http://bicmr.pku.edu.cn/~wyang/132382/notes2.pdf). In page 35, there is a theorem stated that: The limit set of a non-elementary Fuchsian group is the ...
Student's user avatar
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2 votes
1 answer
226 views

Cusp set is dense in boundary of hyperbolic plane

Let $\Gamma$ be a Fuchsian group such that $\mathbb{H}/\Gamma$ is a finite-area hyperbolic surface with some cusps. How can we show that cusp set of $\Gamma$ is dense in the boundary of $\mathbb{H}$? ...
user avatar
1 vote
1 answer
132 views

Dehn's algorithm satisfies linear isoperimetric inequality

A Dehn's presentation for a group is a finite presentation $\langle X; R \rangle$ such that if any non-trivial word $w$ in $F(X)$ represents the identity element of $G$, then there is a relation $r=r_{...
user avatar
2 votes
1 answer
605 views

G a group, H a subgroup of finite index. Proof that G is hyperbolic $\Leftrightarrow$ H is hyperbolic?

I found the the following claim here: If $G$ is a group and $G_0 \subset G$ is a subgroup of finite index, then $G$ is hyperbolic if and only if $G_0$ is hyperbolic. Why is this true? Can anyone ...
wanderingmathematician's user avatar
3 votes
1 answer
133 views

Proof that finite symmetrized relator sets, which are $C'(1/6)$, with equal normal closures are unique

The following statement is made in the Wikipedia article on small cancellation theory without reference or proof. Can anyone either provide a proof or point me to a reference with a proof? The ...
wanderingmathematician's user avatar
3 votes
1 answer
978 views

How to show the free product of two hyperbolic groups is still a hyperbolic group?

I saw from a paper which claimed that this is a easy consequence from the definitions, but I can't give a proof of it just by the definitions. So could you give me some ideas? Thanks!
user avatar
0 votes
1 answer
150 views

Some questions about a proof referring to hyperbolic group

I feel confused about: 1:It says that "otherwise H contains an infinite cyclic characteristic subgroup C," however, by definition, if H is elementary, we can only get that H contains an infinite ...
user avatar
2 votes
1 answer
165 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 ("...
M.U.'s user avatar
  • 1,953
3 votes
0 answers
195 views

Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta)\...
mathreadler's user avatar
2 votes
0 answers
54 views

Deck transformations and Gromov Hyperbolicity

I would like to ask, once more, for some references in Gromov-hyperbolic spaces. The question is specifically the following: Does someone know any alternative reference, alternative proof, anything, ...
matgaio's user avatar
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2 votes
1 answer
214 views

Is every finitely generated Kleinian group commensurable to a Coxeter group?

Or, to a finitely generated reflection group? Here, I do not insist that the Coxeter group is represented as a hyperbolic reflection group. If not, what is an counterexample? And what is a ...
Hao Chen's user avatar
  • 299
2 votes
1 answer
154 views

Infinite geodesic rays leaving a K-quasiconvex subgroup stay K-close to it.

I am going through some basic properties of $\delta$-hyperbolic spaces and groups and I am having some difficulties proving precisey some things that are anyway intuitively clear to me. Let $G$ be a $...
Lor's user avatar
  • 5,587
5 votes
1 answer
474 views

normalizer of a cyclic subgroup in a torsion-free hyperbolic group

How one can show that in a torsion-free hyperbolic group if elements $x$ and $y$ (edit: $y\ne1$) satisfy: $$ xy^mx^{-1}=y^n $$ then $m=n$ and $x$ and $y$ belong to the same cyclic subgroup?
mathreader's user avatar
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