Tits alternative
Linear groups — groups of matrices — turn out to live double lives, but only two of them. Either the group is, in essence, tame and orderly: built up from abelian pieces in a controlled, ‘solvable’ way (after possibly passing to a finite-index subgroup). Or it is wild in a single specific manner: it harbors a free group on two generators, the engine of exponential complexity. There is no third possibility — no middle, intermediate kind of linear group.
The Tits alternative (Jacques Tits, 1972) states: let G be a finitely generated linear group over a field, i.e. a finitely generated subgroup of GL(n, k). Then either G contains a nonabelian free subgroup (free of rank 2), or G is virtually solvable (it has a solvable subgroup of finite index). In characteristic zero the finite-generation hypothesis can be dropped. The free subgroup, when it exists, is produced by a ping-pong argument on a suitable representation.
The dichotomy has sweeping consequences: a linear group is amenable if and only if it is virtually solvable, and a linear group either has exponential growth or is virtually nilpotent (polynomial growth) — there is no intermediate growth among linear groups, in stark contrast with Grigorchuk's nonlinear example. The phrase ‘the Tits alternative holds for a class’ has become a general goal, established for hyperbolic groups, mapping class groups (Ivanov, McCarthy), and Out(F_n) (Bestvina-Feighn-Handel), among others.
SL(2, Z) contains the free group of rank 2 generated by [1, 2; 0, 1] and [1, 0; 2, 1], so it lands on the free side. By contrast the upper-triangular group of matrices [a, b; 0, a^(-1)] is solvable, landing on the virtually-solvable side.
SL(2, Z) has a free subgroup; the triangular group is solvable — the two Tits horns.
‘Virtually P’ means having a finite-index subgroup with property P. The theorem's two horns are not exclusive only by accident: a virtually solvable group cannot contain a nonabelian free subgroup, since free groups of rank 2 are not virtually solvable.