Algebraic actions of discrete groups: the $p$-adic method

Abstract

We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on p-adic analysis, the second makes use of isoperimetric inequalities and LangWeil estimates. For instance, we show that if SL n(Z) acts faithfully on a complex quasi-projective variety X by birational transformations, then dim(X) ≥ n−1 and X is rational if dim(X) = n−1.