K\"ahler groups and subdirect products of surface groups

TL;DR

This paper constructs new classes of K"ahler groups as subgroups of direct products of surface groups, exploring their finiteness properties and constraints, advancing understanding of their geometric and algebraic structure.

Contribution

It introduces a novel construction of K"ahler groups as fibers of maps to tori and characterizes their finiteness properties within direct products of surface groups.

Findings

Constructed infinite classes of irreducible, coabelian K"ahler subgroups of surface group products.

Demonstrated these groups exhibit all possible finiteness properties for their classifying spaces.

Established constraints on K"ahler subdirect products, linking finiteness properties to algebraic structure.

Abstract

We present a construction that produces infinite classes of K\"ahler groups that arise as fundamental groups of fibres of maps to higher dimensional tori. Following the work of Delzant and Gromov, there is great interest in knowing which subgroups of direct products of surface groups are K\"ahler. We apply our construction to obtain new classes of irreducible, coabelian K\"ahler subgroups of direct products of $r$ surface groups. These cover the full range of possible finiteness properties of irreducible subgroups of direct products of $r$ surface groups: For any $r\geq 3$ and $2\leq k \leq r-1$, our classes of subgroups contain K\"ahler groups that have a classifying space with finite $k$-skeleton while not having a classifying space with finitely many $(k+1)$-cells. We also address the converse question of finding constraints on K\"ahler subdirect products of surface groups and, more generally, on homomorphisms from K\"ahler groups to direct products of surface groups. We show that if a K\"ahler subdirect product of $r$ surface groups admits a classifying space with finite $k$-skeleton for $k>\frac{r}{2}$, then it is virtually the kernel of an epimorphism from a direct product of surface groups onto a free abelian group of even rank.