Orderability of big mapping class groups

TL;DR

This paper provides an alternative proof that the big mapping class group of an infinite-type surface with boundary is left-orderable, using inductive constructions of ideal arc systems and topological analysis.

Contribution

It introduces a new inductive method to establish left-orderability of big mapping class groups via generalized ideal arc systems.

Findings

A countable stable Alexander system can be constructed for the surface.

Generalized ideal arc systems induce left-orderings on the big mapping class group.

The topology on the big mapping class group matches the order topology from the constructed left-ordering.

Abstract

We give an alternate proof of the left-orderability of the mapping class group of a connected oriented infinite-type surface with a non-empty boundary. Our main strategy involves the inductive construction of a countable stable Alexander system for the surface using a carefully chosen exhaustion by finite-type subsurfaces. In fact, we prove that a generalised ideal arc system for the surface also induces a left-ordering on the big mapping class group. We then prove that two generalised ideal arc systems determine the same left-ordering if and only if they are loosely isotopic. Finally, we prove that the topology on the big mapping class group is the same as the order topology induced by a left-ordering corresponding to an inductively constructed ideal arc system.