# 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 ...
1 vote
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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 ...
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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 ...
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1 vote
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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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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!
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### 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 ...
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### 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 ("...
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