Divergence of CAT(0) cube complexes and Coxeter groups

Levcovitz, Ivan

doi:10.2140/agt.2018.18.1633

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Divergence of $\mathrm{CAT}(0)$ cube complexes and Coxeter groups

Ivan Levcovitz

Algebraic & Geometric Topology 18 (2018) 1633–1673

DOI: 10.2140/agt.2018.18.1633

Abstract

We provide geometric conditions on a pair of hyperplanes of a $CAT (0)$ cube complex that imply divergence bounds for the cube complex. As an application, we classify all right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani and Thomas that addressed the class of $2$ –dimensional right-angled Coxeter groups. As another application, we provide an inductive graph-theoretic criterion on a right-angled Coxeter group’s defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups that are not right-angled.

Keywords

$\mathrm{CAT}(0)$ cube complex, divergence, Coxeter group, right-angled Coxeter group

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20F55, 57M99

References

Publication

Received: 6 March 2017

Revised: 8 January 2018

Accepted: 25 January 2018

Published: 3 April 2018

Authors

Ivan Levcovitz
Department of Mathematics The Graduate Center, CUNY New York, NY United States
http://www.ivanlevcovitz.com

Volume 18, issue 3 (2018)