On the interior of subsemigroups of Lie groups

Abstract:

Let $G$ denote a Lie group with Lie algebra $\mathfrak {g}$ and with a subsemigroup $S$ whose infinitesimal generators generate $\mathfrak {g}$. We construct real analytic curves $\gamma :{{\mathbf {R}}^ + } \to S$ such that $\dot \gamma (0)$ is a preassigned tangent vector of $S$ at the origin and that $\gamma (t)$ is in the interior of $S$ for all positive $t$. Among the consequences, we find that the boundary of $S$ has to be reasonably well behaved. Our procedure involves the construction of certain linear generating sets from a given Lie algebra generating set, and this may be of independent interest.