Abstract:

We establish general central limit theorems for a general nonelementry action of a wide variety of groups on Gromov hyerbolic spaces with respect to counting for the word length in the group. In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018. In subsequent work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces. Our techniques replace the classical thermodynamic formalism with tools reducing studying balls in the word metric to studying a collection of random walks on the group. We provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. I will discuss generalizing this to representations into higher rank Lie groups with lengths of geodesics replaced by moduli of eigenvalues. Joint work with G. Tiozzo and S. Taylor.

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