Classifying spaces for families of subgroups for systolic groups

TL;DR

This paper explores the large-scale geometry of systolic groups, constructs classifying spaces for virtually cyclic and abelian subgroups, and develops methods for analyzing complexes related to small cancellation and CAT(0) groups.

Contribution

It introduces new constructions of low-dimensional classifying spaces for various subgroup families in systolic and small cancellation groups, linking geometric and algebraic properties.

Findings

Determined the large-scale geometry of minimal displacement sets in systolic complexes.

Constructed low-dimensional classifying spaces for virtually cyclic and abelian subgroups.

Showed graphical small cancellation complexes serve as classifying spaces for proper actions.

Abstract

We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them properly admit three-dimensional classifying spaces with virtually cyclic stabilisers. This is achieved by constructing a systolic complex equivariantly homotopy equivalent to a graphical small cancellation complex. On the way we develop a systematic approach to graphical small cancellation complexes. Finally, we construct low-dimensional models for the family of virtually abelian subgroups for systolic, graphical small cancellation, and some CAT(0) groups.