Automorphisms of the loop and arc graph of an infinite-type surface

TL;DR

This paper establishes isomorphisms between the extended mapping class groups of infinite-type surfaces and automorphism groups of their loop and arc graphs, extending finite-type results to infinite surfaces.

Contribution

It proves that the automorphism group of the loop graph is isomorphic to the extended mapping class group for infinite-type surfaces, extending known finite-type results.

Findings

Automorphism group of the loop graph is isomorphic to the extended mapping class group.

Extended mapping class group stabilizing punctures is isomorphic to the arc graph.

Extends finite-type surface results to infinite-type surfaces.

Abstract

We show that the extended based mapping class group of an infinite-type surface is naturally isomorphic to the automorphism group of the loop graph of that surface. Additionally, we show that the extended mapping class group stabilizing a finite set of punctures is isomorphic to the arc graph relative to that finite set of punctures. This extends a known result for sufficiently complex finite-type surfaces, and provides a new angle from which to study the mapping class groups of infinite-type surfaces.