The deficiencies of Kaehler groups

TL;DR

This paper classifies Kaehler groups with high deficiency, showing limitations on their structure and identifying certain non-Abelian limit groups as surface groups, thus advancing understanding of their algebraic properties.

Contribution

It extends the classification of Kaehler groups by deficiency, proving the non-existence of certain groups and characterizing non-Abelian limit groups as surface groups.

Findings

No Kaehler groups of even positive deficiency exist.

Kaehler non-Abelian limit groups are surface groups.

Classification of Kaehler groups with deficiency at least two.

Abstract

Generalizing the theorem of Green--Lazarsfeld and Gromov, we classify Kaehler groups of deficiency at least two. As a consequence we see that there are no Kaehler groups of even and strictly positive deficiency. With the same arguments we prove that Kaehler groups that are non-Abelian and are limit groups in the sense of Sela are surface groups.