The handlebody group is a virtual duality group

TL;DR

This paper proves that the handlebody mapping class group is a virtual duality group and describes its dualising module via homology of a disc system complex.

Contribution

It establishes the virtual duality property for handlebody mapping class groups and characterizes the dualising module in terms of disc system homology.

Findings

Handlebody mapping class group is a virtual duality group.

Dualising module described as homology of non-simple disc systems.

Applicable to torsion-free, finite-index subgroups.

Abstract

We show that the mapping class group of a handlebody is a virtual duality group, in the sense of Bieri and Eckmann. In positive genus we give a description of the dualising module of any torsion-free, finite-index subgroup of the handlebody mapping class group as the homology of the complex of non-simple disc systems.