  In this thesis we describe a type of metric space called an Euclidean
polyhedral complex. We define a Dirichlet form on it; this is used to give a
corresponding heat kernel. We provide a uniform small time Poincare inequality
for complexes with bounded geometry and use this to determine uniform small
time heat kernel bounds via a theorem of Sturm. We then consider such complexes
with an underlying finitely generated group structure. We use techniques of
Saloff-Coste and Pittet to show a large time asymptotic equivalence for the
heat kernel on the complex and the heat kernel on the group.
