By adapting a technique of Molchanov, we obtain the heat kernel asymptotics
at the sub-Riemannian cut locus, when the cut points are reached by an
r-dimensional parametric family of optimal geodesics. We apply these results
to the bi-Heisenberg group, that is, a nilpotent left-invariant
sub-Rieman\-nian structure on R5 depending on two real parameters
α1​ and α2​. We develop some results about its geodesics and
heat kernel associated to its sub-Laplacian and we illuminate some interesting
geometric and analytic features appearing when one compares the isotropic
(α1​=α2​) and the non-isotropic cases (α1â€‹î€ =α2​). In particular, we give the exact structure of the cut locus, and
we get the complete small-time asymptotics for its heat kernel.Comment: 17 pages, 1 figur