Brownian motion on treebolic space: escape to infinity

摘要

Treebolic space is an analog of the Sol geometry, namely, it is the horocylic product of the hyperbolic upper half plane H and the homogeneous tree T = T-p, with degree p + 1 >= 3, the latter seen as a one-complex. Let h be the Busemann function of T with respect to a fixed boundary point. Then for real q > 1 and integer p >= 2, treebolic space HT(q, p) consists of all pairs (z = x + iy, w) is an element of H x T with h(w) = log(q) y. It can also be obtained by glueing together horizontal strips of Elf in a tree-like fashion. We explain the geometry and metric of HT and exhibit a locally compact group of isometries (a horocyclic product of affine groups) that acts with compact quotient. When q = p, that group contains the amenable Baumslag-Solitar group BS(p) as a co-compact lattice, while when q not equal p, it is amenable, but non-unimodular. HT(q, p) is a key example of a strip complex in the sense of [4].