PUB - Publikationen an der Universität Bielefeld

A linear automorphism of a finite dimensional red vector space V is called proximal if it has a unique eigenvalue-counting multiplicities-of maximal modulus. Goldsheid and Margulis have shown that if a subgroup G of GL(V) contains a proximal element then so does every Zariski dense subsemigroup H of G, provided V considered as a G-module is strongly irreducible. We here show that H contains a finite subset M such that for every g is an element of GL(V) at least one of the elements gamma g, gamma is an element of M, is proximal. We also give extensions and refinements of this result in the following directions: a quantitative version of proximality, reducible representations, several eigenvalues of maximal modulus.