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

TL;DR

This paper investigates the $K$-theory of crossed products from group actions on multitrees, providing explicit formulas and conditions for various properties, advancing understanding of their algebraic and topological structure.

Contribution

It introduces a six-term exact sequence in $K$-theory for crossed products from group actions on multitrees and characterizes properties like minimality and amenability in this context.

Findings

Derived explicit $K$-theory formulas for free and infinite cyclic vertex stabilizer actions.

Established criteria for minimality, local contractivity, and topological freeness of the boundary action.

Extended results to actions on the boundary of undirected trees.

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.