On finitely generated normal subgroups of K\"ahler groups

TL;DR

This paper investigates the structure of K"ahler groups with certain normal subgroups, showing they are often virtually direct products involving surface groups, and providing restrictions on more complex subgroup embeddings.

Contribution

It establishes new conditions under which normal subgroups of K"ahler groups are virtually surface groups or direct products, extending understanding of their algebraic structure.

Findings

Surface groups embedded as normal subgroups imply the K"ahler group is virtually a product with a surface group.

Hyperbolic groups with infinite outer automorphism groups embedded normally are virtually surface groups.

Restrictions are given on normal subgroups that are amalgamated products or HNN extensions.

Abstract

We prove that if a surface group embeds as a normal subgroup in a K\"ahler group and the conjugation action of the K\"ahler group on the surface group preserves the conjugacy class of a non-trivial element, then the K\"ahler group is virtually given by a direct product, where one factor is a surface group. Moreover we prove that if a one-ended hyperbolic group with infinite outer automorphism group embeds as a normal subgroup in a K\"ahler group then it is virtually a surface group. More generally we give restrictions on normal subgroups of K\"ahler groups which are amalgamated products or HNN extensions.