Presqu'un immeuble pour le groupe des automorphismes mod\'er\'es

TL;DR

This paper constructs a geometric space related to the tame automorphism group of affine space, showing it has non-positive curvature in certain cases, and uses this to prove linearizability of finite subgroups.

Contribution

It introduces a new CAT(0) space associated with Tame($K^n$) and establishes its geometric properties, providing insights into the structure of automorphism groups.

Findings

X is a Euclidean CW-complex of dimension n-1

For n=3 and characteristic zero, X is a CAT(0) space

Finite subgroups of Tame($K^3$) are linearizable

Abstract

Inspired by the Bruhat-Tits building of SL$_n$($\mathbb Q_p$), we construct a complete metric space X with an action of the tame automorphism group of the affine space Tame($K^n$). The points in X are certain monomial valuations, and X admits a natural structure of Euclidean CW-complex of dimension n-1. When n = 3, and for K of characteristic zero, we prove that X has non-positive curvature and is simply connected, hence is a CAT(0) space. As an application we obtain the linearizability of finite subgroups in Tame($K^3$).