## Acta Mathematica

### 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 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 $\mathrm{dim}(X) = n-1$.

#### Article information

Source
Acta Math., Volume 220, Number 2 (2018), 239-295.

Dates
Revised: 3 February 2018
First available in Project Euclid: 19 June 2019

https://projecteuclid.org/euclid.acta/1560966692

Digital Object Identifier
doi:10.4310/ACTA.2018.v220.n2.a2

Mathematical Reviews number (MathSciNet)
MR3849285

Zentralblatt MATH identifier
06925265

#### Citation

Cantat, Serge; Xie, Junyi. Algebraic actions of discrete groups: the $p$-adic method. Acta Math. 220 (2018), no. 2, 239--295. doi:10.4310/ACTA.2018.v220.n2.a2. https://projecteuclid.org/euclid.acta/1560966692