Abstract

The celebrated notion of hyperbolic graphs was invented by M. Gromov in 1987 for developing geometric group theory. In the area of analysis on fractals and metric measure spaces, the hyperbolic graphs of the main interest turn out to be those arising from iterations, partitions or ball coverings. In this seminar, we formulated a broad class of hyperbolic graphs (called expansive hyperbolic graphs), and investigated the hyperbolic bound-arizations of compact metric spaces in a wide fractal-like setup. To develop a discrete potential theory on compact doubling spaces, we constructed a class of reversible random walks on certain hyperbolic graphs such that the Martin boundaries are homeomorphic to the underlying spaces, and the (Sil-verstein's) induced energy forms can be analyzed. This is based on some joint work with Ka-Sing Lau (CUHK), Xiang-Yang Wang (SYSU) and Ting-Kam Leonard Wong (Toronto).