Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actions

TL;DR

This paper develops an equivariant Morse theory framework applied to Vietoris-Rips complexes, enabling the construction of finite universal spaces for proper group actions in the context of asymptotically CAT(0) groups.

Contribution

It introduces an equivariant version of Bestvina-Brady Morse theory and applies it to produce finite universal spaces for proper actions for a broad class of groups.

Findings

Established an equivariant Morse theory framework

Constructed finite universal spaces for asymptotically CAT(0) groups

Extended Morse theory techniques to Vietoris-Rips complexes

Abstract

We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.