Les groupes de Burger-Mozes ne sont pas k\"ahl\'eriens

TL;DR

This paper proves that Burger-Mozes groups, constructed as lattices in automorphism groups of cubical buildings, cannot be the image of a K"ahler group via a morphism with finitely generated kernel, thus they are not K"ahler.

Contribution

It establishes that Burger-Mozes groups are not K"ahler by showing no morphism from a K"ahler group with finitely generated kernel exists.

Findings

Burger-Mozes groups are not K"ahler.

No morphism from a K"ahler group with finitely generated kernel to these groups exists.

These groups cannot be realized as K"ahler groups.

Abstract

Burger and Mozes constructed examples of infinite simple groups which are lattices in the group of automorphisms of a cubical building. We show that there can be no morphism with finitely generated kernel from a K\"ahler group to one of these groups. We obtain as a consequence that these groups are not K\"ahler.---Burger et Mozes ont construit des exemples de groupes simples infinis, qui sont des r\'eseaux dans le groupe des automorphismes d'un immeuble cubique. On montre qu'il n'existe pas de morphisme d'un groupe k\"ahl\'erien vers l'un de ces groupes dont le noyau soit finiment engendr\'e. On en d\'eduit que ces groupes ne sont pas k\"ahl\'eriens.