Counting loxodromics for actions of hyperbolic groups and other automatic groups
主 题: Counting loxodromics for actions of hyperbolic groups and other automatic groups
报告人: Ilya Gekhtman (Yale University)
时 间: 2018-03-21 14:30-16:30
地 点: Room 1365, Sciences Building No.1
Abstract: We show that for arbitrary nonelementary actions $G\curvearrowright X$ of hyperbolic groups on Gromov hyperbolic spaces, translation length on average grows linearly in word length. In particular, the proportion of loxodromic elements in a large ball in the Cayley graph converges to 1. This holds even when the action is not in any sense alignment preserving: for example a dense free subgroup of $SL_2R$ acting on the hyperbolic plane, or a hyperbolic subgroup of the mapping class group acting on the curve complex. Along the way we described the behavior in the space $X$ of typical word geodesics in the group: for example, with respect to the Patterson-Sullivan measure on the boundary group, the orbit of almost every word geodesic logarithmically tracks a geodesic in $X$. We prove analogous counting results for more general groups, including relatively hyperbolic groups with virtually abelian subgroups and right angled Artin and Coxeter groups. Our results hold more generally for automatic groups satisfying certain properties: groups parametrized by paths in a finite directed graph. Indeed, the automatic structure is what allows us to reduce the asymptotic geometry of the Cayley graph of $G$ to a certain Markov chain on a finite graph and a family? of random walks on $G$ associated to vertices of the finite graph. This is joint work with Sam Taylor and Giulio Tiozzo.
