The term linear group refers to a group which is isomorphic to a subgroup of the general linear group GL(n, F) for some natural number n and some field F. The basic result, settling the growth problem for linear groups, is the following

Theorem 6.1 (The Tits Alternative – [Ti 72]) Let G be a finitely generated linear group. Then either G contains a non-abelian free subgroup, or G contains a soluble subgroup of finite index.

Corollary 6.2The growth of a finitely generated linear group is either exponential or polynomial, and it is polynomial if and only if the group is nilpotent-by-finite.

Indeed, if the group contains a non-abelian free subgroup its growth is exponential. In the other case, apply Corollary 5.4.

Y. Shalom [Sh 98] gave a proof of Corollary 6.2 independent of the Tits' alternative for the case of linear groups of characteristic zero. This case suffices for the proof of Gromov's theorem described below. It was proved by A. Eskin, S. Mozes, and Hee Oh [EMO 05], that linear groups of characteristic zero of exponential growth are of uniformly exponential growth, and E. Breuillard and T. Gelander [BG 08] extended this to all characteristics. This is done by showing that if G is not nilpotent-byfinite, then we can find a non-abelian free subsemigroup of G, for which the lengths of the free generators are bounded, relative to whatever set of generators of G is taken.