Emmanuel BREUILLARD On the Tits Alternative and its Refinements

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I shall present Tits' original statement and method of proof of his celebrated alternative for finitely generated linear groups and present various efinements that have been obtained over the years as well as open problems. In particular, I shall discuss the notion of strongly dense free subgroup appearing in recent joint work with Guralnick and Larsen on this topic as well as connections to the study of word maps and expander 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