Tree Pairs for Algebraic Bieri-Strebel Groups

TL;DR

This paper revisits a method for representing Algebraic Bieri-Strebel groups using tree pairs and shows that some higher order groups lack such representations, indicating no universal degree bound for these groups.

Contribution

It reintroduces a known tree pair construction method and identifies classes of groups that cannot be represented by tree pairs, expanding understanding of their limitations.

Findings

Certain higher degree groups lack tree pair representations

No maximum polynomial degree guarantees a tree pair representation

Reintroduction of a method for constructing tree pair representations

Abstract

We reintroduce a previously discovered method for constructing tree pair representations for Algebraic Bieri-Strebel groups, as well as demonstrate a class of higher order groups that cannot have a tree pair representation. In doing so, we demonstrate that there is no maximum degree such that for all polynomials of higher degree, the associated Algebraic Bieri Strebel group must have a tree-pair representation.