Branched covers of elliptic curves and K\"ahler groups with exotic finiteness properties

TL;DR

This paper constructs K"ahler groups with customizable finiteness properties by holomorphically mapping products of Riemann surfaces onto elliptic curves, revealing new exotic behaviors in their classifying spaces.

Contribution

It introduces a method to generate K"ahler groups with specific finiteness properties, expanding understanding of their topological and geometric complexity.

Findings

Constructed K"ahler groups with arbitrary finiteness properties.

Identified invariants distinguishing these groups.

Inspired by and extending previous examples by Dimca, Papadima, and Suciu.

Abstract

We construct K\"ahler groups with arbitrary finiteness properties by mapping products of closed Riemann surfaces holomorphically onto an elliptic curve: for each $r\geq 3$, we obtain large classes of K\"ahler groups that have classifying spaces with finite $(r-1)$-skeleton but do not have classifying spaces with finitely many $r$-cells. We describe invariants which distinguish many of these groups. Our construction is inspired by examples of Dimca, Papadima and Suciu.