Contractible Rips complexes of groups via metric gluings

Abstract

We say that a finitely generated group equipped with a finite generating set is of type $\mathsf{R}$ if the Rips complex is contractible for sufficiently large scales. We show that the class of groups of type $\mathsf{R}$ is closed under taking graphs of groups with finite edge groups and, in particular, under taking free products. We prove that all two-dimensional right-angled Artin groups with the standard generating set are of type $\mathsf{R}$. Furthermore, we observe that products of finitely generated free abelian groups and finite groups are of type $\mathsf{R}$.

9 pages