On James Hyde's example of non-orderable subgroup of $\mathrm{Homeo}(D,\partial D)$

TL;DR

The paper revisits Hyde's example of a non-left-orderable subgroup of disk homeomorphisms, providing a new dynamical proof and extending the analysis to actions on the circle.

Contribution

It offers a new dynamical proof of Hyde's example and extends the non-orderability result to actions on the circle.

Findings

New dynamical proof of Hyde's non-orderable subgroup

Extension of results to actions on the circle

Demonstration that $ ext{Homeo}(D,oundary D)$ is not left-orderable

Abstract

In [Ann. Math. 190 (2019), 657-661], James Hyde presented the first example of non-left-orderable, finitely generated subgroup of $\mathrm{Homeo}(D,\partial D)$, the group of homeomorphisms of the disk fixing the boundary. This implies that the group $\mathrm{Homeo}(D,\partial D)$ itself is not left-orderable. We revisit the construction, and present a slightly different proof of purely dynamical flavor, avoiding direct references to properties of left-orders. Our approach allows to solve the analogue problem for actions on the circle.