# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.00407