Orderable Thompson-like groups arising from Ore categories
Davide Perego, Matteo Tarocchi
TL;DR
The paper addresses the problem of orderability for fundamental groups of Thompson-like categories built as right-Ore indirect products $\mathcal{F}\bowtie\mathcal{G}$. It develops a general method using $\Phi$-discrete orderings on $\mathcal{G}$ compatible with a left- or bi-order on $\mathcal{F}$ to deduce that $\Pi(\mathcal{F}\bowtie\mathcal{G})$ is left- or bi-orderable, respectively. The authors apply this framework to digit rewriting systems, showing that groups of $\mathcal{G}$-fractions are left-orderable while purely braided fractions are bi-orderable, and derive new results such as the left-orderability of braided Houghton groups. The approach recovers known Thompson-like outcomes (Higman–Thompson, Houghton, and topological full groups) and yields new braided generalizations with clear implications for algebraic and dynamical properties of these groups.
Abstract
We give sufficient conditions for left- and bi-orderability of fundamental groups of Ore categories in terms of indirect factors, including Thompson groups and many of their generalizations. Besides recovering known results, we prove that braided groups of fractions of digit rewriting systems (which generalize braided Thompson groups to the wider setting of topological full groups of edge shift) are left-orderable, and that their purely braided counterparts are bi-orderable. In particular, the braided Houghton groups are left-orderable.