Abstract
We prove that, for any irreducible Artin-Tits group of spherical type G, the quotient of G by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on G, and constructing a specific element \(x_G\) of G / Z(G) whose action on the graph is loxodromic and WPD in the sense of Bestvina–Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that “generic” elements of G act loxodromically, where the word “generic” can be understood in either of the two common usages: as a result of a long random walk or as a random element in a large ball in the Cayley graph.
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Acknowledgements
The first author was supported by the “initiation to research” Project No. 11140090 from Fondecyt and by Dr. Andrés Navas through the Project USA1555 from University of Santiago de Chile. He also acknowledges support by PIA-CONICYT ACT1415 and by MTM2010-19355 and FEDER.
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Calvez, M., Wiest, B. Acylindrical hyperbolicity and Artin-Tits groups of spherical type. Geom Dedicata 191, 199–215 (2017). https://doi.org/10.1007/s10711-017-0252-y
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DOI: https://doi.org/10.1007/s10711-017-0252-y