Lihuang Ding

Dynamics, Geometry and Groups Seminar

Speaker: Lihuang Ding

Title: Word-Metric Counting Problems in Acylindrically Hyperbolic Groups

Abstract:
Counting in word-metric balls is a dynamical problem with a different sampling law from a random walk. For an isometric action $G$ on $X$, the orbit map and the stable translation length record the asymptotic motion of an element. This talk explains a counting analogue of linear progress for acylindrically hyperbolic groups: generic elements make linear progress in an auxiliary hyperbolic space and have almost maximal stable word and translation lengths. The geometric input contains an action with independent loxodromic WPD elements in the hyperbolic space. The translates of the axes are organized into a projection complex, a hyperbolic quasi-tree. Anchored word-length estimates then produce growth gaps. A linearly recurrence dichotomy then yields exponential genericity of WPD elements for every finite generating set. The talk then describes the consequences: infinite normal quotients have strictly smaller exponential growth and satisfy a strict cogrowth bound, while the number of conjugacy classes intersectinging a word ball is comparable to $|B_n|/n$. We explain the common geometric mechanism linking these results. This talk is based on joint work with Wenyuan Yang."