2023/11/09 by Nathan Brownlowe, Brownlowe, Nathan, Jack Spielberg +5
Computer Science · Mathematics · #20E08 (Secondary) #46L80 (Primary) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2311.05285
openalex publication_date 2023/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C0(∂ E)\rtimesr 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 C0(∂ E) \rtimesr 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 ∂ 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.