Thickness, relative hyperbolicity, and randomness in Coxeter groups. With an appendix written jointly with Pierre-Emmanuel Caprace.
(2017) Algebraic & Geometric Topology — Vol. 17, p. 705-740 (2017)
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- For the finite simplicial graph Γ let WΓ be the corresponding right-angled Coxeter group. The authors prove that every right-angled Coxeter group either is thick or else admits a canonical relatively hyperbolic structure in which the peripheral subgroups are thick. Let T be the class of finite simplicial graphs whose corresponding right-angled Coxeter groups are strongly algebraically thick. It is shown that T is the smallest class of graphs satisfying several graph-theoretic conditions. As a consequence there is a polynomial time algorithm to decide whether a given graph is in T. The authors establish the asymptotic probability that a random right-angled Coxeter group is thick. In an appendix all Coxeter groups are analyzed. Each such group either is strongly algebraically thick or admits a minimal relatively hyperbolic structure. The notion intrinsic horosphericity is introduced, which provides a dynamical obstruction to relative hyperbolicity which generalizes thickness.
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Behrstock, J., Hagen, M., Sisto, A., & Caprace, P.-E. (2017). Thickness, relative hyperbolicity, and randomness in Coxeter groups. With an appendix written jointly with Pierre-Emmanuel Caprace. Algebraic & Geometric Topology, 17, 705-740. https://hdl.handle.net/2078.5/27349 (Original work published 2017)