Bohr compactifications of groups and rings

TL;DR

This paper develops a model-theoretic framework for understanding the Bohr compactifications of rings and groups, providing explicit calculations for classical matrix groups and introducing new concepts of connected components.

Contribution

It introduces model-theoretic connected components of rings and applies them to explicitly compute Bohr compactifications of various classical matrix groups.

Findings

Explicit formulas for Bohr compactifications of matrix groups.

Development of a theory of connected components for rings.

Application to classical groups like the Heisenberg group.

Abstract

We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group $UT_3(Z)$, the continuous Heisenberg group $UT_3(R)$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.