Thickness, relative hyperbolicity, and randomness in Coxeter groups

@article{Behrstock2013ThicknessRH,
  title={Thickness, relative hyperbolicity, and randomness in Coxeter groups},
  author={Jason A. Behrstock and Mark F. Hagen and Alessandro Sisto and Pierre‐Emmanuel Caprace},
  journal={Algebraic \& Geometric Topology},
  year={2013},
  volume={17},
  pages={705-740}
}
For right-angled Coxeter groups $W_{\Gamma}$, we obtain a condition on $\Gamma$ that is necessary and sufficient to ensure that $W_{\Gamma}$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, the peripheral subgroups are both parabolic (in the Coxeter group-theoretic sense) and strongly algebraically thick. We exhibit a polynomial-time algorithm that… 

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