Constructions of $E_{\mathcal{VC}}$ and $E_{\mathcal{FBC}}$ for groups acting on $\mathrm{CAT}(0)$ spaces

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 $E_{\mathcal{V}\mathcal{C}}\Gamma$ and $E_{\mathcal{ℱ}\mathcal{ℬ}\mathcal{C}}\Gamma$ 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 $E_{\mathcal{V}\mathcal{C}}\Gamma$ for crystallographic groups $\Gamma$.