Orderings of mapping class groups after Thurston

TL;DR

This paper explores how Thurston's hyperbolic structure method can generate and classify various left orderings of mapping class groups of surfaces with boundary, revealing finiteness in certain classes.

Contribution

It introduces a natural Thurston-based construction for ordering mapping class groups and classifies all orderings of braid groups arising from this approach.

Findings

Classifies all orderings of braid groups from Thurston's method

Proves finiteness of conjugacy classes for certain nonpathological orderings

Provides a new perspective on ordering structures in surface mapping class groups

Abstract

We are concerned with mapping class groups of surfaces with nonempty boundary. We present a very natural method, due to Thurston, of finding many different left orderings of such groups. The construction involves equipping the surface with a hyperbolic structure, embedding the universal cover in the hyperbolic plane, and extending the action of the mapping class group on it to its limit points on the circle at infinity. We classify all orderings of braid groups which arise in this way. Moreover, restricting to a certain class of ``nonpathological'' orderings, we prove that there are only finitely many conjugacy classes of such orderings.