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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

If Γ is a group acting properly by semisimple isometries on a proper CAT(0) space X, then we build models for the classifying spaces EVCΓ and EℱℬCΓ under the additional assumption that the action of Γ 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 Γ 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 EVCΓ for crystallographic groups Γ.

$\mathrm{CAT}(0)$ space, classifying space, virtually cyclic group
Mathematical Subject Classification 2000
Primary: 18F25, 55N15
Secondary: 20F65
Received: 14 February 2009
Revised: 31 August 2010
Accepted: 2 September 2010
Published: 30 October 2010
Daniel Farley
Department of Mathematics
Miami University
Room 123 Bachelor Hall
Oxford OH 45056