It is well known fact that a (f.g.) group is hyperbolic if and only if it admits a (finite) Dehn presentation.


My question concerns something I'm struggling with since the first time I read the proof of this Theorem (in Bridson, Haefliger, Metric spaces of non-positive curvature). Are there a lot of known examples of (hyperbolic) groups with Dehn presentations (except trivial ones, like free groups)?


It seems (at least to me) that "no one" wants to "really compute" such presentations explicitly. (I know there are implemented algorithms for computing hyperbolic constants of hyperbolic groups)


The secondary reason for my question may be quite silly. I was talking to friend of mine (also mathematician) about this beautiful proof and he asked me: "Can you give me some examples?" .... I was stunned.



Explicit examples of Dehn presentations of hyperbolic groups
I'll turn YCor's comment (above) into an answer. Surface groups are the simplest (and historically first) non-free example. Let $S\_g$ be the surface of genus $g$. Then the fundamental group has presentation



> 
> $\pi\_1(S\_g) \cong \langle a\_1, a\_2, \ldots, a\_{2g-1}, a\_{2g} \mid a\_1 a\_2 \ldots a\_{2g} A\_1 A\_2 \ldots A\_{2g} \rangle$.
> 
> 
> 


This presentation is a Dehn presentation. A variant of this fact is due to Dehn. See the following [post](https://mathoverflow.net/questions/136948/dehns-algorithm-for-word-problem-for-surface-groups) for discussion and references. 


