Classifying spaces for families of subgroups of 8-located groups

TL;DR

This paper studies the structure of minimal displacement sets in 8-located complexes with the SD'-property, showing they embed isometrically and are systolic, leading to a construction of classifying spaces for virtually cyclic subgroups.

Contribution

It introduces a new understanding of minimal displacement sets in 8-located complexes and constructs low-dimensional classifying spaces for certain group actions.

Findings

Minimal displacement sets embed isometrically into the complex.

Such sets are systolic under certain conditions.

Constructs low-dimensional classifying spaces for virtually cyclic subgroups.

Abstract

We investigate the structure of the minimal displacement set in $8$-located complexes with the SD'-property. We show that such set embeds isometrically into the complex. Since $8$-location and simple connectivity imply Gromov hyperbolicity, the minimal displacement set in such complex is systolic. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on an $8$-located complex with the SD'-property.