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

Abstract

We show that the extended based mapping class group of an infinitetype 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.

Big mapping class groups: an overview

“…This is in stark contrast to Map(S 2 C), which Calegari shows admits no quasimorphisms (and even stronger, we know Map(S 2 C) is CB [84]). We note that the automorphism group of A ∞ and related graphs are computed in [110] and shown to be the extended mapping class group.…”