Distality in Ordered Abelian Groups

TL;DR

This paper characterizes when ordered abelian groups are distal based on the uniform boundedness of rib sizes related to prime-specific valuations, extending previous criteria for groups with finite spines.

Contribution

It provides a new characterization of distality in ordered abelian groups using prime-specific valuation properties, generalizing earlier finite spine results.

Findings

Distality is characterized by uniform bounds on rib sizes for each prime p.

The criterion extends previous results for groups with finite spines.

Provides a complete characterization of distality in this class of groups.

Abstract

We provide a characterization of distal ordered abelian groups: An ordered abelian group is distal if and only if, for each prime number $p$, the sizes of ribs with respect to the "valuation" $\mathfrak{s}_p$ are uniformly bounded. This generalizes the distality criterion for ordered abelian groups with finite spines given by Aschenbrenner, Chernikov, Gehret, and Ziegler.