Group actions on multitrees and the 𝐾-theory of their crossed products

Brownlowe, Nathan; Spielberg, Jack; Thomas, Anne; Wu, Victor

doi:10.1090/tran/9304

Group actions on multitrees and the $K$-theory of their crossed products
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by Nathan Brownlowe, Jack Spielberg, Anne Thomas and Victor Wu;

Trans. Amer. Math. Soc. 378 (2025), 1697-1732

DOI: https://doi.org/10.1090/tran/9304

Published electronically: December 13, 2024

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Abstract:

We study group actions on multitrees, which are directed graphs in which there is at most one directed path between any two vertices. In our main result we describe a six-term exact sequence in $K$-theory for the reduced crossed product $C_0(\partial E)\rtimes _r G$ induced from the action of a countable discrete group $G$ on a row-finite, finitely-aligned multitree $E$ with no sources. We provide formulas for the $K$-theory of $C_0(\partial E) \rtimes _r G$ in the case where $G$ acts freely on $E$, and in the case where all vertex stabilisers are infinite cyclic. We study the action $G\curvearrowright \partial E$ in a range of settings, and describe minimality, local contractivity, topological freeness, and amenability in terms of properties of the underlying data. In an application of our main theorem, we describe a six-term exact sequence in $K$-theory for the crossed product induced from a group acting on the boundary of an undirected tree.

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Group actions on multitrees and the $K$-theory of their crossed products
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Abstract: