Dynamics of isolated orders

TL;DR

This paper investigates the properties of isolated left orders in countable groups, demonstrating uniqueness of minimal sets, finiteness of convex subgroups, providing a dynamical proof of Tararin's theorem, and introducing a new isolated order on B_3.

Contribution

It offers new insights into the structure of isolated left orders, including a dynamical proof of Tararin's theorem and the discovery of a new isolated order on the braid group B_3.

Findings

Unique minimal set for dynamical realization if G is not infinite cyclic

Finiteness of convex subgroups of an isolated left order

Existence of a new isolated order on the braid group B_3

Abstract

We study some properties of the dynamical realization of isolated left orders of a countable group G. We show that the dynamical realization admits a unique minimal set provided G is not infinite cyclic. We show that convex subgroup of an isolated left order is finite in number. Using this, we give a dynamical proof of the Tararin theorem. We also show that there is a new isolated order on the braid group B_3.