On the classifying space of the family of finite and of virtually cyclic subgroups for CAT(0)-groups

TL;DR

This paper constructs low-dimensional models for classifying spaces of proper and virtually cyclic subgroups of CAT(0)-groups, based on the group's action on a CAT(0)-space with bounded topological dimension.

Contribution

It establishes the existence of finite-dimensional G-CW-models for classifying spaces of proper and virtually cyclic subgroups of CAT(0)-groups, with explicit dimension bounds.

Findings

Existence of G-CW-model E_fin(G) with dim ≤ d.

Existence of G-CW-model E_vcyc(G) with dim ≤ d+1 under cocompactness.

Dimension bounds depend on the topological dimension of the CAT(0)-space.

Abstract

Let G be a discrete group which acts properly and isometrically on a complete CAT(0)-space X. Consider an integer d with d=1 or d greater or equal to 3 such that the topological dimension of X is bounded by d. We show the existence of a G-CW-model E_fin(G) for the classifying space for proper G-actions with dim(E_fin(G)) less or equal to d. Provided that the action is also cocompact, we prove the existence of a G-CW-model E_vcyc(G) for the classifying space of the family of virtually cyclic subgroups such that dim(E_vcyc(G)) is less or equal to d+1.