Connectivity of partial basis complexes of freely decomposable groups

TL;DR

This paper proves that the complex of partial bases of free groups, viewed up to conjugation, is Cohen--Macaulay of dimension n-1, confirming a conjecture and extending results to freely decomposable groups.

Contribution

It establishes the Cohen--Macaulay property of the partial basis complex for free and freely decomposable groups, confirming a conjecture by Day and Putman.

Findings

The complex is Cohen--Macaulay of dimension n-1.

The result confirms a conjecture by Day and Putman.

The proof extends to freely decomposable groups.

Abstract

We show that the complex of partial bases of the free group of rank $n$, where vertices are seen up to conjugation, is Cohen--Macaulay of dimension $n-1$. This positively answers a conjecture raised by Day and Putman. We prove our results in the more general context of freely decomposable groups.