Tits alternative

The Tits alternative states that every finitely generated linear group either has a solvable subgroup of finite index or contains a nonabelian free subgroup.

Linear group: A group realized as a subgroup of invertible matrices over a field. The theorem applies to finitely generated groups in this broad class.
Finitely generated group: A group generated by a finite set of elements. Finite generation is an explicit hypothesis of the theorem.
Jacques Tits: A Belgian-born French mathematician known for foundational work in group theory and geometry. He proved the theorem that bears his name.
Free subgroup: A subgroup isomorphic to a free group. The theorem guarantees such a subgroup whenever its solvable alternative fails.
Ping-pong lemma: A method for proving that group elements generate a free subgroup by tracking their actions on a set. It is a common technique for establishing the free-subgroup outcome.
Virtually solvable group: A group with a solvable subgroup of finite index. This is one of the theorem’s two possible outcomes.
Field (mathematics): A set with addition, subtraction, multiplication, and division by nonzero elements satisfying the field axioms. Linear groups are formed from matrices over a field.
Burnside problem: The question of whether finitely generated groups of bounded exponent must be finite. It helped frame questions about how finite generation constrains group structure.
Polynomial growth: Growth in which the number of elements within radius n is bounded by a polynomial in n. Virtually solvable linear groups have constrained growth behavior.
Alternative for mapping class groups: A theorem stating that subgroups of mapping class groups are virtually abelian or contain nonabelian free groups. It has a similar dichotomy but applies to mapping-class-group subgroups.

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