The Compression Semigroup of a Cone is Connected

Joao Ribeiro Gonçalves Filho and Luiz A.B. San Martin

Abstract: Let $W\subset\R^{n}$ 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 $\mathrm{Sl}(n,\R)$ are classified into types, each type being labelled by a flag manifold. The semigroups whose type is given by the projective space $\P^{n-1}$ form one of the classes. It is proved here that the semigroups in $\mathrm{Sl}(n,\R)$ leaving invariant a pointed and generating cone are the only maximal connected in the class of $\P^{n-1}$.

Keywords: semigroups; convex cones; positive matrices; maximal connected semigroups.

Classification (MSC2000): 20M20, 11C20.

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