The growth of linear groups
In this nice short paper, the author gives a fairly simple proof of the growth dichotomy for finitely generated linear groups over fields of characteristic zero, namely that each such group has either polynomial or exponential growth. The proof avoids using the Tits alternative (every finitely generated linear group is virtually soluble or contains a free group of rank two). Indeed, the author claims that his main theorem (which follows easily from Tits' result) can replace the Tits alternative in many contexts (e.g.\ in Gromov's proof that every finitely generated group of polynomial growth is virtually nilpotent). The main theorem of the paper says that a finitely generated linear (in characteristic zero) and amenable group is virtually soluble (i.e., it has a soluble subgroup of finite index). From this the growth dichotomy readily follows since any group of subexponential growth is amenable and the dichotomy holds for every virtually soluble group by a theorem due to Wolf and Milnor.
scientific article; zbMATH DE number 1115988