The Alexander method for infinite-type surfaces

TL;DR

This paper establishes a set of essential curves on any infinite-type orientable surface that uniquely determines the identity homeomorphism, proving the faithfulness of the mapping class group's action on the curve complex.

Contribution

It introduces a countable collection of curves for infinite-type surfaces that characterizes the identity homeomorphism and demonstrates the faithful action of the extended mapping class group.

Findings

Existence of a countable essential curve collection for infinite-type surfaces.

Homeomorphisms preserving these curves are isotopic to the identity.

The extended mapping class group's action on the curve complex is faithful.

Abstract

We prove that for any infinite-type orientable surface S there exists a collection of essential curves {\Gamma} in S such that any homeomorphism that preserves the isotopy classes of the elements of {\Gamma} is isotopic to the identity. The collection {\Gamma} is countable and has infinite complement in C(S), the curve complex of S. As a consequence we obtain that the natural action of the extended mapping class group of S on C(S) is faithful.