Given a $\delta$-hyperbolic group $G$, I have been told that the Rips $n$-complex of $G$ is contractible for high enough $n$. The only proof I have found for this statement is in an [expository essay](https://warwick.ac.uk/fac/sci/maths/people/staff/lopezdegamiz/hyperbolic_groups.pdf) by Jone Lopez de Gamiz. I don't believe the proof there is independent, but I can't tell where it's from. Can someone direct me to a more original source for this theorem?



Looking for a citation: the Rips $n$-complex of a $\delta$-hyperbolic group is contractible for high enough $n$
In his monograph *Hyperbolic groups* (1987), Gromov states and proves:


**Lemma 1.7.A.** *Let $X$ be a $\delta$-hyperbolic space
such that every $x\in X$ can be joined by a segment with
a fixed reference point $x\_0 \in X$. Then the polyhedron
$P\_d(X)$ is contractible for all $d \geq 4 \delta$.*


Here, the polyhedron $P\_d(X)$ is the Rips complex whose simplices are the finite collections of points in $X$ that are pairwise at distance $\leq d$.


This is probably the first published reference where the statement appears, even though the result is attributed to E. Rips. In the late 1980s, several texts have been dedicated to hyperbolic groups in general, and the the result can be found there too:


* E. Ghys, P. de la Harpe, *Sur les groupes hyperboliques d'après Mikhael Gromov* (1989).
* E. Ghys, *Les groupes hyperboliques*, [Séminaire Bourbaki](http://perso.ens-lyon.fr/ghys/articles/groupeshyperboliquesbourbaki.pdf) (1989-1990).
* Coornaert, Delzant, Papadopoulos, *Géométrie et théorie des groupes* (1990).
* E. Ghys, A. Haefliger, A. Verjovsky, *Group theory from a geometrical viewpoint* (1991).


But the result can also be found in more modern references, such Bridson and Haefliger's book *Metric spaces of non-positive curvature*, as already mentioned; or Drutu and Kapovich's book *Geometric group theory*.


