Cohen--Macaulay Complexes, Duality Groups, and the dualizing module of ${\rm{Out}}(F_N)$

TL;DR

This paper explores the relationship between Cohen--Macaulay complexes and duality properties in groups, providing a new description of the dualizing module of ${\rm{Out}}(F_N)$ using local cohomology of Outer space.

Contribution

It establishes a connection between Cohen--Macaulay classifying spaces and Bieri--Eckmann duality, and describes the dualizing module of ${\rm{Out}}(F_N)$ in terms of local cohomology.

Findings

Cohen--Macaulay classifying spaces are common among groups satisfying Bieri--Eckmann duality.

A comparison between Bieri--Eckmann duality and Cohen--Macaulay complex duality is provided.

The dualizing module of ${\rm{Out}}(F_N)$ is characterized via local cohomology of Outer space.

Abstract

We explain how Cohen--Macaulay classifying spaces are ubiquitous among discrete groups that satisfy Bieri--Eckmann duality, and compare Bieri--Eckmann duality to duality results for Cohen--Macaulay complexes. We use this comparison to give a description of the dualizing module of ${\rm{Out}}(F_N)$ in terms of the local cohomology cosheaf of the spine of Outer space.