Tits alternative

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

Statement

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.

Generalization

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).

References

• Tits, J. (1972). "Free subgroups in linear groups". Journal of Algebra 20 (2): 250–270. doi:10.1016/0021-8693(72)90058-0.
• Bestvina, Mladen; Feighn, Mark; Handel, Michael (2000). "The Tits alternative for Out(F_n) I: Dynamics of exponentially-growing automorphisms". Annals of Mathematics 151 (2): 517–623. arXiv:math/9712217. doi:10.2307/121043. JSTOR 121043.