I heard that the Rips complexes associated to the Cayley graphs of hyperbolic groups are contractible for a sufficiently large radius. Is the converse true? Namely, if a group is non-hyperbolic, then is its Rips complex never ``asymptotically" contractible?


For example, we can ask if the non-hyperbolic group $\mathbb{Z}^2$ has contractible Rips complex associated to its Cayley graph for large radius.


I don't know much geometric group theory, and quick Google searches for this question weren't enough to fetch me an easy answer for this. This seemed like a natural question that would have been addressed at some point, so if anyone has a standard answer for this, it would be nice to know :)



Contractible Rips complex from non-hyperbolic group
Another source of Cayley graphs with contractible Rips complexes comes from Helly graphs.


**Proposition:** *Rips complexes of uniformly locally finite Helly graphs are contractible.*


See Lemma 5.28 and Theorem 4.2(v) from the preprint [arXiv:2002.06895](https://arxiv.org/abs/2002.06895).


One construction of Helly graphs is the following: Given a CAT(0) cube complex $X$, the graph obtained from $X^{(1)}$ by adding an edge between any two vertices which belong to a common cube is a Helly graph. And there exist many groups admitting one-skeleta of CAT(0) cube complexes as Cayley graphs. For instance:


**Corollary:** *Let $\Gamma$ be a finite simplicial graph. Let $G$ denote the Cayley graph of the right-angled Artin group $A(\Gamma)$ constructed from the generating set $\{ u\_1\cdots u\_n \mid u\_1, \ldots, u\_n \in V(\Gamma) \text{ pairwise adjacent} \}$. Then all the Rips complexes of $G$ are contractible.*


Notice that $A(\Gamma)$ is hyperbolic if and only if $\Gamma$ has no edges, so most of these examples are not hyperbolic. For instance, the corollary includes $\mathbb{Z}^2$ with the generating set $\{(1,0), (1,1), (0,1)\}$. Other groups having one-skeleta of CAT(0) cube complexes as Cayley graphs include right-angled Coxeter groups and right-angled mock reflection groups.


**Edit:** I should also mention [arxiv:1904.09060](https://arxiv.org/abs/1904.09060), which proves that Garside groups (such as braid groups) and Artin groups of type FC also admit Cayley graphs that are Helly, and so whose Rips complexes are contractible.


