Constructions of E𝒱𝒞 and Eℱℬ𝒞 for groups acting on CAT(0) spaces

Farley, Daniel

doi:10.2140/agt.2010.10.2229

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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

DOI: 10.2140/agt.2010.10.2229

Abstract

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 $E_{V C} Γ$ and $E_{ℱ ℬ C} Γ$ 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 $E_{V C} Γ$ for crystallographic groups $Γ$ .

Keywords

$\mathrm{CAT}(0)$ space, classifying space, virtually cyclic group

Mathematical Subject Classification 2000

Primary: 18F25, 55N15

Secondary: 20F65

References

Publication

Received: 14 February 2009

Revised: 31 August 2010

Accepted: 2 September 2010

Published: 30 October 2010

Authors

Daniel Farley
Department of Mathematics Miami University Room 123 Bachelor Hall Oxford OH 45056 USA
http://www.users.muohio.edu/farleyds/

Volume 10, issue 4 (2010)