Cubulable K\"ahler groups

TL;DR

This paper characterizes K"ahler groups that act properly and cocompactly on CAT(0) cubical complexes, showing they are virtually products of surface groups and free Abelian groups, and describes the structure of related K"ahler manifolds.

Contribution

It establishes a structural classification of cubulable K"ahler groups and their associated manifolds, revealing their finite index subgroups and geometric decompositions.

Findings

K"ahler groups acting on CAT(0) cubical complexes are virtually products of surface groups and free Abelian groups.

A closed aspherical K"ahler manifold with a cubulable fundamental group admits a finite cover as a torus bundle over Riemann surfaces.

A factorization result for essential actions of K"ahler groups on irreducible CAT(0) cubical complexes is proved.

Abstract

We prove that a K\"ahler group which is cubulable, i.e. which acts properly discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index subgroup isomorphic to a direct product of surface groups, possibly with a free Abelian factor. Similarly, we prove that a closed aspherical K\"ahler manifold with a cubulable fundamental group has a finite cover which is biholomorphic to a topologically trivial principal torus bundle over a product of Riemann surfaces. Along the way, we prove a factorization result for essential actions of K\"ahler groups on irreducible, locally finite CAT(0) cubical complexes, under the assumption that there is no fixed point in the visual boundary.