2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119440We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative.Group Theory20G25 ; 22E40text