Block Conjugacy of Irreducible Toral Automorphisms

TL;DR

This paper explores the concept of block conjugacy in irreducible toral automorphisms, establishing its equivalence to ideal weak equivalence and analyzing conditions for conjugacy and Galois group relations.

Contribution

It introduces block conjugacy for irreducible toral automorphisms and links it to ideal theory, Galois groups, and subgroup actions, providing new characterizations.

Findings

Block conjugacy is equivalent to weak ideal equivalence.

Characterization of conjugacy via group actions on subtori.

Analysis of relationships between non-block conjugate automorphisms.

Abstract

We introduce a relation of block conjugacy for irreducible toral automorphism, and prove that block conjugacy is equivalent to weak equivalence of the ideals associated to the automorphisms. We characterize when block conjugate automorphisms are actually conjugate in terms of a group action on invariant and invariantly complemented subtori, and detail the relation of block conjugacy with a Galois group. We also investigate the nature of the relationship between ideals associated to non-block conjugate irreducible automorphisms.