The compression semigroup of a cone is connected

Let $W\subset \Bbb{R}^{n}$ be a pointed and generating cone and denote by $S(W)$ the semigroup of matrices with positive determinant leaving $W$ invariant. The purpose of this paper is to prove that $S(W)$ is path connected. This result has the following consequence: Semigroups with nonempty interior in the group $\mathrm{Sl}( n,\Bbb{R}) $ are classified into types, each type being labelled by a flag manifold. The semigroups whose type is given by the projective space $\Bbb{P}^{n-1}$ form one of the classes. It is proved here that the semigroups in $\mathrm{Sl}(n,\Bbb{R})$ leaving invariant a pointed and generating cone are the only maximal connected in the class of $\Bbb{P}^{n-1}$.