Questions tagged [hyperbolic-groups]

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0answers
224 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 ...
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0answers
25 views

Loxodromic element of an hyperbolic group

I struggle to show the following thing : Let $G$ be an $\delta$-hyperbolic group related to the space X, $g$ be a loxodromic element. Then $E(g)=\lbrace u \in G | \exists \in \mathbb{N}, ~ ug^ku^{-1}=...
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1answer
66 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 ...
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1answer
81 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^{+} = \{ ...
1
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1answer
47 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 ...
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1answer
133 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 ...
2
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1answer
90 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 $\mathbb{H}$? ...
1
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1answer
66 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_{...
1
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1answer
205 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 ...
3
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1answer
76 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 ...
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1answer
317 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!
0
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1answer
79 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 ...
2
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1answer
85 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 ("...
3
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0answers
128 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)\...
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0answers
45 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, ...
1
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1answer
127 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 ...
2
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1answer
109 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 $...
4
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1answer
290 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?