JournalsrmiVol. 31, No. 3pp. 935–976

Brownian motion on treebolic space: escape to infinity

  • Alexander Bendikov

    Uniwersytet Wrocławski, Poland
  • Laurent Saloff-Coste

    Cornell University, Ithaca, United States
  • Maura Salvatori

    Università di Milano, Italy
  • Wolfgang Woess

    Technische Universität Graz, Austria
Brownian motion on treebolic space: escape to infinity cover
Download PDF


Treebolic space is an analog of the Sol\mathsf {Sol} geometry, namely, it is the horocylic product of the hyperbolic upper half plane H\mathbb H and the homogeneous tree T=Tp\mathbb T=\mathbb T_{\mathsf p} with degree p+13\mathsf p+1 \ge 3, the latter seen as a one-complex. Let h\mathfrak h be the Busemann function of T\mathbb T with respect to a fixed boundary point. Then for real q>1\mathsf q > 1 and integer p2\mathsf p \ge 2, treebolic space HT(q,p)\mathsf {HT}(\mathsf q,\mathsf p) consists of all pairs (z=x+iy,w)H×T(z=x+\mathfrak i y,w) \in \mathbb H \times \mathbb T with h(w)=logqy\mathfrak h (w) = \mathrm {log}_{\mathsf q} y. It can also be obtained by glueing together horizontal strips of H\mathbb H in a tree-like fashion. We explain the geometry and metric of HT\mathsf HT and exhibit a locally compact group of isometries (a horocyclic product of affine groups) that acts with compact quotient. When q=p\mathsf q=\mathsf p, that group contains the amenable Baumslag–Solitar group BSp)\mathsf {BS} \mathsf p) as a co-compact lattice, while when qp\mathsf q \ne \mathsf p, it is amenable, but non-unimodular. HT(q,p)\mathsf {HT} (\mathsf q,\mathsf p) is a key example of a strip complex in the sense of [4].$

Relying on the analysis of strip complexes developed by the same authors in [4], we consider a family of natural Laplacians with "vertical drift" and describe the associated Brownian motion. The main difficulties come from the singularities which treebolic space (as any strip complex) has along its bifurcation lines. In this first part, we obtain the rate of escape and a central limit theorem, and describe how Brownian motion converges to the natural geometric boundary at infinity. Forthcoming work will be dedicated to positive harmonic functions.

Cite this article

Alexander Bendikov, Laurent Saloff-Coste, Maura Salvatori, Wolfgang Woess, Brownian motion on treebolic space: escape to infinity. Rev. Mat. Iberoam. 31 (2015), no. 3, pp. 935–976

DOI 10.4171/RMI/859