On Kahler extensions of abelian groups

TL;DR

This paper characterizes Kahler extensions of finitely generated abelian groups by surface groups, showing they are virtually products, and explores conditions under which such extensions arise from homomorphisms into mapping class groups.

Contribution

It establishes a classification of Kahler extensions involving abelian and surface groups and links these to homomorphisms into mapping class groups.

Findings

Kahler extensions are virtually products under certain conditions

Homomorphisms into mapping class groups can produce Kahler extensions

Restrictions from surface topology limit Kahler group structures

Abstract

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. The main tools come from surface topology and known restrictions on Kahler groups.