Positive cones and bi-orderings on almost-direct products of free groups

TL;DR

This paper provides an explicit description of positive cones that define bi-invariant orderings on almost-direct products of free groups, with applications to braid groups and automorphism groups.

Contribution

It introduces a new explicit construction of positive cones using normal forms and Magnus-type orderings, clarifying the structure of bi-invariant orderings on these groups.

Findings

Explicit positive cones are compatible with natural projections.

Canonical subgroups are convex within the order structure.

The construction applies to pure monomial braid groups and McCool groups.

Abstract

Almost-direct products of free groups arise naturally in braid theory and in the study of automorphism groups of free groups. Although bi-invariant orderings are known to exist for many such groups, their explicit structure is often left implicit. In this paper, we give an explicit description of the positive cones defining bi-invariant orderings on almost-direct products of free groups, using normal forms derived from the almost-direct product decomposition together with Magnus-type orderings on free factors. We establish key structural properties of these cones, including compatibility with natural projections, convexity of canonical subgroups, and invariance under suitable classes of automorphisms. As applications, we show how the construction applies to several families of groups of geometric and algebraic interest, such as pure monomial braid groups and McCool groups.