The Tits Alternative

We shall say that the Tits Alternative holds for a class of groups if each group in the class is either solvable by ...nite (that is, contains a solvable normal subgroup of ...nite index) or contains a free subgroup of rank at least 2. Free groups of rank 2 contain free subgroups of countable rank, so in the second case there will be of free subgroups of all ...nite ranks. The Tits Alternative is so called because it arises in a major theorem published by J. Tits [20] in 1972: TITS’ THEOREM Let G be a ...nitely generated linear group over a (commutative) ...eld. Then eitherG is solvable by ...nite orG contains a noncyclic free subgroup. ¤ The object of the present paper is to give an exposition of a simpli...ed version of Tits’ proof, and to discuss some of the rami...cations of Tits’ Theorem. In what follows F will denote an arbitrary (commutative) ...eld, and a linear group of degree n over F will refer to a subgroup of GL(n;F ), or a group of linear transformations of an n-dimensional vector space over F . Acknowledgements The present paper is a major revision of [5] published in 1972, which purports to give a proof of Tits’ Theorem but contains some very serious ‡aws. I am indebted to several mathematicians, including B. Wehrfritz and Ju.I. Merzljakov, who pointed out what now appear as embarrassingly elementary errors, and I am particularly indebted to J. Hulse who in his correspondence in 1974 insisted that I get it right. Why then should I publish this revised version at this late date? The reasons are two-fold. Firstly, as far as I know, except for the original paper of Tits the only places where the proof of his theorem appears are in [14] and [23] both of which follow the original proof quite closely. (Actually, only an outline of the proof appears in [23]). The proof given below in Sects. 3-4 seems (at least to me) to be technically simpler, partly because Tits is interested in proving a more general result and some of the technicalities can be avoided for that reason. Since the theorem remains of major interest, it seems worthwhile having an alternative approach to its proof. Secondly, it seems worthwhile putting on record in Sect. 2 some of the work which has arisen as a consequence of the theorem its applications and some generalizations which it has stimulated. The reference journal Citation Index records over 60 citations to Tits’ original paper up to the end of 1987.