Title: The Manhattan curve and rough similarity rigidity
Speaker: Ryokichi Tanaka – Kyoto University
Abstract: For every non-elementary hyperbolic group, we consider the Manhattan curve, which was originally introduced by M. Burger (1993),
associated with any pair of (say) word metrics. It is convex; we show that it is continuously differentiable and moreover is a straight line if and only if the corresponding two metrics are roughly similar, that is, they are within bounded distance after multiplying by a positive constant. I would like to explain how it is related to the central limit theorem for uniform counting measures on spheres, to ergodic theory of topological flows built on general hyperbolic groups, and to the multifractal structure of Patterson-Sullivan measures. Furthermore, I will present some explicit examples including a hyperbolic triangle group and compute the exact value of the mean distortion for a pair of word metrics by using automatic structures of the group.
Joint work with Stephen Cantrell (University of Chicago).
Zoom link: https://osu.zoom.us/j/91638927725
Meeting ID: 916 3892 7725
Password: Mixing
Recorded Talk: https://osu.zoom.us/rec/share/LPAREOkf7DJ5zT6tKMPY1dZP_Vs824-zAEFnAJ7CTzXJKS-pMk_Bx8o2eD6QXg-y.lFp9Na68FCmloePw