Connectivity of partial basis complexes of 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.

arXiv comment12 pages; v2: accepted version with improved exposition and further background in the introduction; to appear in Bull. Lond. Math. Soc

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