Let ∆ = −

Abstract. Let G be a connected Lie group of polynomial growth. We consider m-th order subelliptic differential operators H on G, the semigroups St = e −tH and the corresponding heat kernels Kt. For a large class of H with m> 4 we demonstrate equivalence between the existence of Gaussian bounds on Kt, with “good ” large t behaviour, and the existence of “cutoff” functions on G. By results of [14], such cutoff functions exist if and only if G is the local direct product of a compact Lie group and a nilpotent Lie group