  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 with
degree p+1 > 2, 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 > 1, treebolic space HT(q,p) consists of all pairs (z=x+i y,w) in H x
T with h(w) = log_{q} y. It can also be obtained by glueing together horziontal
strips of H 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 and p are distinct, it is amenable, but non-unimodular. HT(q,p) is a key
example of a strip complex in the sense of our previous paper in Advances in
Mathematics 226 (2011) 992-1055.
  Relying on the analysis of strip complexes developed in that paper, we
consider a family of natural Laplacians with "vertical drift" and describe the
associated Brownian motion. The main difficulties come from the singularites
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.

Probability;Differential Geometry;Group Theory