Tits alternative

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In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.


Every finitely generated linear group is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a subgroup isomorphic to the free group on two generators.


In geometric group theory, a group G is said to satisfy the Tits alternative if for every subgroup H of G either H is virtually solvable or H contains a nonabelian free subgroup (in some versions of the definition this condition is only required to be satisfied for all finitely generated subgroups of G).