Generalized Bowen-Franks Groups and Profinite Conjugacy for Hyperbolic Toral Automorphisms

TL;DR

This paper introduces generalized Bowen-Franks groups as complete invariants for profinite conjugacy of hyperbolic toral automorphisms, expanding the understanding of their classification.

Contribution

It establishes that principal Bowen-Franks R-modules fully characterize profinite conjugacy for similar hyperbolic toral automorphisms, providing a new invariant framework.

Findings

Principal Bowen-Franks R-modules are complete invariants.

These modules are the principal invariants in a broad class of invariants.

Cyclicity invariants are realized for profinite conjugacy.

Abstract

We show that a collection of generalized Bowen-Franks group, what we call the principal Bowen-Franks $R$-modules, form a complete set of $R$-module invariants for the equivalence relation of profinite conjugacy for similar hyperbolic toral automorphisms. We also show that these principal generalized Bowen-Franks $R$-modules are the principal invariants in a large class of generalized Bowen-Franks $R$-module invariants for similar hyperbolic automorphisms. We further show how the conjugacy invariant of cyclicity, when applied to hyperbolic toral automorphisms, is realized for profinite conjugacy.