Kodaira fibrations, K\"ahler groups, and finiteness properties

TL;DR

This paper constructs new K"ahler groups from Kodaira fibrations that lack finite classifying spaces and are not related to surface group products, revealing novel properties of these complex geometric groups.

Contribution

It introduces a new class of K"ahler groups with unique finiteness properties, expanding understanding of their algebraic and geometric structures.

Findings

Constructed K"ahler groups without finite classifying spaces

Demonstrated these groups are not commensurable to subdirect products of surface groups

Linked these groups to fundamental groups of fibers in holomorphic maps from Kodaira fibrations

Abstract

We construct classes of K\"ahler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map from a product of Kodaira fibrations onto an elliptic curve.