Contractibility of Vietoris–Rips complexes for groups with contractible geometric models

Determine whether every finitely generated group admitting a proper cocompact action on a contractible simplicial complex has a finite parameter t for which its Vietoris–Rips complex VR_t(G), with respect to some finite generating set and the associated word metric, is contractible.

Background

The paper studies contractibility and high connectivity of Vietoris–Rips complexes of finitely generated groups equipped with word metrics. Hyperbolic groups provide a major positive example, but the authors note that comparatively little is known for groups outside the hyperbolic setting. A necessary condition is that the group act properly and cocompactly on some contractible complex, yet the paper states that this condition is not known to be sufficient.

The question concerns both the existence of a finite generating set and the existence of a finite Vietoris–Rips parameter t. The paper also notes a stronger unresolved variant asking whether such a parameter exists for every finite generating set, but the broader existence question is the clearest explicitly stated open problem.

References

Of course a necessary condition is that $G$ act geometrically on a contractible complex, but little else is known. Even the seemingly easy example of $Zn$ ($n\ge 2$) with the standard generating set was only recently handled, by Virk in , and in general the problem is wide open.

\begin{question} Let $G$ be a finitely generated group that admits a proper cocompact action on some contractible simplicial complex. Must some $VR_t(G)$ ($t<\infty$) be contractible? \end{question}

— Word length, Morse theory, and Vietoris-Rips complexes  (2608.25614 - Hulbert et al., 26 Aug 2026) in Section 1, Introduction, Question environment