【摘要】
In this talk, we will introduce the Martin boundary of groups - a compactification arising from random walks on groups that completely characterizes all non-negative harmonic functions on groups. We focus on the connections between this probability-related boundary and various geometric boundaries of groups.
We begin by reviewing some results in hyperbolic and relatively hyperbolic groups. Then we introduce Ancona inequalities, which describe the multiplicative behavior of Green functions along geodesics. We establish Ancona inequalities for Morse subsets under specific geometric conditions. A key result we present is the construction of an injective mapping from a full-measure subset of the Roller boundary to the Martin boundary for some right-angled Coxeter groups. For general groups with contracting elements, most aspects of this construction remain valid. This is based on a joint work with Wenyuan Yang.
