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 $\mathrm{dim}(X) = n-1$.
Citation
Serge Cantat. Junyi Xie. "Algebraic actions of discrete groups: the $p$-adic method." Acta Math. 220 (2) 239 - 295, June 2018. https://doi.org/10.4310/ACTA.2018.v220.n2.a2