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

Ivan Levcovitz

Algebraic & Geometric Topology 18 (2018) 1633–1673

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.

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$\mathrm{CAT}(0)$ cube complex, divergence, Coxeter group, right-angled Coxeter group
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F55, 57M99
Received: 6 March 2017
Revised: 8 January 2018
Accepted: 25 January 2018
Published: 3 April 2018
Ivan Levcovitz
Department of Mathematics
The Graduate Center, CUNY
New York, NY
United States