  We introduce the notion of strip complex. A strip complex is a special type
of complex obtained by gluing "strips" along their natural boundaries according
to a given graph structure. The most familiar example is the one dimensional
complex classically associated with a graph, in which case the strips are
simply copies of the unit interval (our setup actually allows for variable edge
length). A leading key example is treebolic space, a geometric object studied
in a number of recent articles, which arises as a horocyclic product of a
metric tree with the hyperbolic plane. In this case, the graph is a regular
tree, the strips are the closed unit interval times the real line, and each
strip is equipped with the hyperbolic geometry of a specific strip in upper
half plane. We consider natural families of Dirichlet forms on a general strip
complex and show that the associated heat kernels and harmonic functions have
very strong smoothness properties. We study questions such as essential
selfadjointness of the underlying differential operator acting on a suitable
space of smooth functions satisfying a Kirchoff type condition at points where
the strip complex bifurcates. Compatibility with projections that arise from
proper group actions is also considered.
