Generalized ideal classes in application to toroidal solenoids

Maria Sabitova

Vol. 318 (2022), No. 1, 189–228

DOI: 10.2140/pjm.2022.318.189

Abstract

We study certain subgroups $G_A$ of ${\mathbb{Q}}^n$ defined by nonsingular $n\times n$-matrices $A$ with integer coefficients. In the first nontrivial case when $n=2$, we give necessary and sufficient conditions for two such groups to be isomorphic. Namely, in the generic case when the characteristic polynomial of $A$ is irreducible, we attach a generalized ideal class to $A$ and essentially, two groups are isomorphic if and only if the corresponding ideal classes are equivalent. The obtained results can be applied to studying associated toroidal solenoids.

Keywords

ideal classes, solenoids, odometers, matrices

Mathematical Subject Classification

Primary: 11Z05

Milestones

Received: 25 June 2021

Revised: 11 November 2021

Accepted: 2 April 2022

Published: 1 August 2022