Acta Mathematica

Volume 220 (2018)

Number 2

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

Pages: 239 – 295

DOI: http://dx.doi.org/10.4310/ACTA.2018.v220.n2.a2

Authors

Serge Cantat (Institut de recherche mathématique, Université de Rennes 1, France)

Junyi Xie (Institut de recherche mathématique, Université de Rennes 1, France)

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 Lang–Weil estimates. For instance, we show that, if $\mathsf{SL}_n(\mathbf{Z})$ acts faithfully on a complex quasi-projective variety $X$ by birational transformations, then $\mathrm{dim}(X) \geqslant n-1$ and $X$ is rational if $\mathrm{dim}(X) = n-1$.

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Received 7 July 2015

Received revised 3 February 2018