On ordered groups of regular growth rates

TL;DR

This work introduces growth order groups as an axiom-based framework for comparing regular growth rates under composition across diverse settings, including o-minimal germs, transseries, and non-commutative series. It constructs and analyzes these groups via a triad of core ingredients: growth axioms $\ref{gog1}$–$\ref{gog3}$, a non-commutative valuation with a convex centralisers structure, and the notion of scaling elements and skeletons. It then develops concrete construction techniques (semidirect products, quotients, finite-value sets) and links to H-fields with composition and inversion, using Taylor expansions to establish when germ groups form growth order groups with Archimedean centralisers. The results unify growth-rate phenomena in Hardy and o-minimal contexts, provide a robust toolkit for embedding and decomposing such groups, and yield new insights into the first-order and structural properties of these asymptotic groups.

Abstract

We introduce an elementary class of linearly ordered groups, called growth order groups, encompassing certain groups under composition of formal series (e.g. transseries) as well as certain groups $\mathcal{G}_{\mathcal{M}}$ of infinitely large germs at infinity of unary functions definable in an o-minimal structure $\mathcal{M}$. We study the algebraic structure of growth order groups and give methods for constructing examples. We show that if $\mathcal{M}$ expands the real ordered field and germs in $\mathcal{G}_{\mathcal{M}}$ are levelled in the sense of Marker & Miller, then $\mathcal{G}_{\mathcal{M}}$ is a growth order group.

Theorems & Definitions (130)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1.1
• Remark 1
• Lemma 1.2
• Corollary 1.3
• Corollary 1.4
• proof
• Proposition 2.1