P-adic lattices are not K\"ahler groups

TL;DR

This paper proves that lattices in simple p-adic Lie groups cannot serve as fundamental groups of compact Kähler manifolds, extending the understanding of the relationship between p-adic groups and geometric structures.

Contribution

It establishes a new non-Kählerness result for lattices in simple p-adic Lie groups, filling a gap in the classification of Kähler groups.

Findings

Lattices in simple p-adic Lie groups are not Kähler groups.

The result applies to certain variants of these lattices.

This advances the understanding of which groups can be Kähler.

Abstract

In this note we show that any lattice in a simple p-adic Lie group is not the fundamental group of a compact Ka\"hler manifold, as well as some variants of this result.