Homology of odometers

Abstract

We compute the homology groups of transformation groupoids associated with odometers and show that certain infinite dihedral group odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_r^{\ast }(G)$. We also show that transformation grupoids of odometers satisfy the AH conjecture.

arXiv comment10 pages. Typos corrected and details added

References in corpus (6)

• $G$-odometers and their almost 1-1 extensions
• Cocycle superrigidity for profinite actions of irreducible lattices
• On Virtual Conjugacy of Generalized Odometers
• Ample groupoids: equivalence, homology, and Matui's HK conjecture
• Groupes pleins-topologiques [d'après Matui, Juschenko, Monod...]
• C*-algebras associated to graphs of groups

Cited by in corpus (6)

• Homology and K-theory of dynamical systems. I. torsion-free ample groupoids
• Nuclear dimension of subhomogeneous twisted groupoid C*-algebras and dynamic asymptotic dimension
• Homology and K-theory of dynamical systems. IV. Further structural results on groupoid homology
• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• Katsura--Exel--Pardo Groupoids and the AH~Conjecture
• Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces