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 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?
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asked Jun 7, 2022 at 18:44
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6$\begingroup$ There is a proof in Bridson–Haefliger as Proposition III.$\Gamma$.3.23 $\endgroup$– Rylee LymanJun 7, 2022 at 19:14
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1$\begingroup$ Unfortunately, the Bridson–Haefliger result doesn't seem to cite an older paper? $\endgroup$– Rylee LymanJun 7, 2022 at 19:15
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1$\begingroup$ I believe it is also in the book by Ghys and de la Harpe. $\endgroup$– Benjamin SteinbergJun 7, 2022 at 23:13
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2$\begingroup$ @BenjaminSteinberg: You're right, this is Item 10 on page 72. $\endgroup$– AGenevoisJun 8, 2022 at 4:36
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2$\begingroup$ Chapter 5 in Cooarnert, Delzant, and Papadopoulos' book Géométrie et théorie des groupes is dedicated to Rips complexes of hyperbolic spaces. $\endgroup$– AGenevoisJun 8, 2022 at 4:39
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1 Answer
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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 (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.
