Kähler groups and subdirect products of surface groups

Llosa Isenrich, Claudio

doi:10.2140/gt.2020.24.971

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Kähler groups and subdirect products of surface groups

Claudio Llosa Isenrich

Geometry & Topology 24 (2020) 971–1017

DOI: 10.2140/gt.2020.24.971

Abstract

We present a construction that produces infinite classes of Kähler 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 Kähler. We apply our construction to obtain new classes of irreducible, coabelian Kähler 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 Kähler 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 Kähler subdirect products of surface groups and, more generally, on homomorphisms from Kähler groups to direct products of surface groups. We show that if a Kähler 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.

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Keywords

Kähler groups, surface groups, branched covers, homological finiteness properties

Mathematical Subject Classification 2010

Primary: 20F65, 32J27

Secondary: 20J05, 32Q15

References

Publication

Received: 26 June 2018

Revised: 10 June 2019

Accepted: 30 August 2019

Published: 23 September 2020

Proposed: Walter Neumann

Seconded: Benson Farb, Anna Wienhard

Authors

Claudio Llosa Isenrich
Max Planck Institute for Mathematics Bonn Germany	Faculty of Mathematics University of Vienna Vienna Austria
https://www.mat.univie.ac.at/~isenrich/

Volume 24, issue 2 (2020)