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Abstract

Let $G$ be a Lie group of polynomial volume growth. Consider a differential operator $H$ of order $2m$ on $G$ which is a sum of even powers of a generating list $A_1, \ldots, A_{d'}$ of right invariant vector fields. When $G$ is solvable, we obtain an algebraic condition on the list $A_1, \ldots, A_{d'}$ which is sufficient to ensure that the semigroup kernel of $H$ satisfies global Gaussian estimates for all times. For $G$ not necessarily solvable, we state an analytic condition on the list which is necessary and sufficient for global Gaussian estimates. Our results extend previously known results for nilpotent groups.