The AH conjecture for Cantor minimal dihedral systems

TL;DR

This paper proves the AH conjecture for transformation groupoids arising from minimal actions of the infinite dihedral group on the Cantor set, linking homology groups with the abelianization of the topological full group.

Contribution

It demonstrates that these specific groupoids satisfy the AH conjecture, using Kakutani-Rokhlin partitions tailored to the systems.

Findings

The AH conjecture holds for minimal dihedral group actions on the Cantor set.

The proof employs adapted Kakutani-Rokhlin partitions.

This extends the class of systems known to satisfy the conjecture.

Abstract

The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani-Rokhlin partitions adapted to such systems.