Algebraic & Geometric Topology

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

Daniel Farley

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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 ECΓ 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.

Article information

Algebr. Geom. Topol., Volume 10, Number 4 (2010), 2229-2250.

Received: 14 February 2009
Revised: 31 August 2010
Accepted: 2 September 2010
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18F25: Algebraic $K$-theory and L-theory [See also 11Exx, 11R70, 11S70, 12- XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67] 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

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


Farley, Daniel. Constructions of $E_{\mathcal{VC}}$ and $E_{\mathcal{FBC}}$ for groups acting on $\mathrm{CAT}(0)$ spaces. Algebr. Geom. Topol. 10 (2010), no. 4, 2229--2250. doi:10.2140/agt.2010.10.2229.

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