The Growth of Linear Groups

Let G be a group generated by a finite subset S; define S to be the set Ž . < n < of all products of at most n elements of S, and let a S s S be the n n Ž . Ž . Ž . Ž . number of elements in S . As a S satisfies 1 F a S F a S ? a S , n nqm n m Ž .1r n Ž . Ž .1r n the limit lim a S exists, and a S s lim a S G 1. Although the n n Ž . exact value of a S depends on the generating set S, it is well known and Ž . Ž . easy to verify that whether a S ) 1 or a S s 1, depends only on the group G. In the first case we say that G has exponential growth, whereas in the second, that G has subexponential growth. If there exist constants c, d Ž . d such that a S F cn for all n, then G is said to have polynomial growth, n Ž and again, this property depends only on the group itself for details see w x. Pa, Chap. 6 . Obviously, if G has polynomial growth it has also subexponential growth. The first construction of a group having the latter property, w x but not the former one, was given in Gri . In this note we prove that every finitely generated linear group over a Ž field of characteristic 0 has either polynomial or exponential growth we . refer to this phenomenon as the ‘‘growth dichotomy’’ . This result was w x obtained by Tits Ti as a consequence of his remarkable ‘‘alternative theorem,’’ but apparently no simple proof for it has been given. Our