Constructions of E_{vc} and E_{fbc} for groups acting on CAT(0) spaces

TL;DR

This paper constructs models for classifying spaces E_{vc} and E_{fbc} for groups acting on CAT(0) spaces, assuming well-behaved axes, extending previous work on crystallographic groups.

Contribution

It introduces a new method to build classifying spaces for groups acting on CAT(0) spaces with specific axis properties, broadening the scope of known constructions.

Findings

Models for E_{vc} and E_{fbc} are constructed under new assumptions.

The approach generalizes previous constructions for crystallographic groups.

Conjecture that the axis condition holds in many cases.

Abstract

If G is a group acting properly by semisimple isometries on a proper CAT(0) space X, then we build models for the classifying spaces E_{vc} and E_{fbc} under the additional assumption that the action of G has a well-behaved collection of axes in X. (This hypothesis is described in the paper.) We conjecture that the latter hypothesis is satisfied in a large range of cases. Our classifying spaces resemble those created by Connolly, Fehrman, and Hartglass for crystallographic groups G.