Acylindrical hyperbolicity and Artin-Tits groups of spherical type. (English) Zbl 1423.20028
Summary: 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 M. Bestvina and K. Fujiwara [Geom. Topol. 6, 69–89 (2002; Zbl 1021.57001)]; following D. Osin [Trans. Am. Math. Soc. 368, No. 2, 851–888 (2016; Zbl 1380.20048)], 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.
MSC:
| 20F36 | Braid groups; Artin groups |
| 20F65 | Geometric group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 57M07 | Topological methods in group theory |
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