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.

中文翻译：

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离散群的代数作用：$p$-adic 方法

我们研究准射影变体的自同构群和双有理变换群。两种方法相结合；第一个基于 p-adic 分析，第二个使用等周不等式和 LangWeil 估计。例如，我们证明如果 SL n(Z) 通过双有理变换忠实地作用于复杂的拟射影变体 X，那么如果 dim(X) = n-1，则 dim(X) ≥ n−1 且 X 是有理数。