Genus 3 Mapping Class Groups are not Kahler

TL;DR

This paper demonstrates that certain finite index subgroups of genus 3 mapping class and Torelli groups are not Kahler, based on explicit algebraic presentations showing their unipotent completions are not quadratic.

Contribution

It provides explicit presentations of unipotent completions for genus 3 Torelli and mapping class groups, revealing their non-quadratic nature and extending results to related groups.

Findings

Genus 3 Torelli groups are not Kahler due to non-quadratic presentations.

Presentations for higher genus Torelli and mapping class groups are quadratic.

Groups related to hyperelliptic and low genus mapping class groups are not Kahler.

Abstract

In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera > 3; they are quadratic. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera < 3 are not Kahler.