#### Volume 10, issue 4 (2010)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Constructions of $E_{\mathcal{VC}}$ and $E_{\mathcal{FBC}}$ for groups acting on $\mathrm{CAT}(0)$ spaces

### Daniel Farley

Algebraic & Geometric Topology 10 (2010) 2229–2250
##### Abstract

If $\Gamma$ is a group acting properly by semisimple isometries on a proper $CAT\left(0\right)$ space $X$, then we build models for the classifying spaces ${E}_{\mathsc{V}\mathsc{C}}\Gamma$ and ${E}_{\mathsc{ℱ}\mathsc{ℬ}\mathsc{C}}\Gamma$ under the additional assumption that the action of $\Gamma$ has a well-behaved collection of axes in $X$. We verify that the latter assumption is satisfied in two cases: (i) when $X$ has isolated flats, and (ii) when $X$ is a simply connected real analytic manifold of nonpositive sectional curvature. We conjecture that $\Gamma$ has a well-behaved collection of axes in the great majority of cases.

Our classifying spaces are natural variations of the constructions due to Connolly, Fehrman and Hartglass [arXiv:math.AT/0610387] of ${E}_{\mathsc{V}\mathsc{C}}\Gamma$ for crystallographic groups $\Gamma$.

##### Keywords
$\mathrm{CAT}(0)$ space, classifying space, virtually cyclic group
##### Mathematical Subject Classification 2000
Primary: 18F25, 55N15
Secondary: 20F65