An Alexander method for infinite-type surfaces

TL;DR

This paper extends the Alexander method to infinite-type surfaces, providing new tools for understanding their mapping class groups and applications to subgroup centers and topological bases.

Contribution

The paper introduces an extension of the Alexander method specifically for infinite-type surfaces, advancing the analysis of their mapping class groups.

Findings

Verified a relation in the mapping class group

Showed centers of many twist subgroups are trivial

Provided a smaller basis for the topology of the mapping class group

Abstract

The Alexander method is a combinatorial tool used to determine when two elements of the mapping class group are equal. We extend the Alexander method to include the case of infinite-type surfaces. Versions of the Alexander method was proven by Hern\'andez--Morales--Valdez and Hern\'andez--Hidber. As sample applications, we verify a relation in the mapping class group, show that the centers of many twist subgroups of the mapping class group are trivial, and provide a relatively smaller basis for the topology of the mapping class group.