THE ASYMPTOTIC SEMIGROUP OF A SUBSEMIGROUP OF A NILPOTENT LIE GROUP

Abstract

Let S be a subsemigroup of a simply connected nilpotent Lie group G. We construct an asymptotic semigroup S0 in the associated graded Lie group G0 of G. We can compute the image of S0 in the abelianization {G}_0^{mathrm{ab}}={G}^{mathrm{ab}}. This gives useful information about S. As an application, we obtain a transparent proof of the following result of E. B. Vinberg and the author: either there is an epimorphism f : G → ℝ such that f (s) ≥ 0 for every s in S or the closure overline{S} of S is a subgroup of G and G/overline{S} is compact.