  We develop a general framework for spectral gap inequalities for Gibbs
measures on infinite dimensional spin spaces over nilpotent Lie groups in terms
of weak U-bounds and weak single-site spectral gap inequalities. We then
provide sufficient conditions on the local specification and give examples of
measures constructed using the Kaplan norm and generalising a few results for
the Carnot-Caratheodory distance on the Heisenberg group.
