Algebraic and topological properties of big mapping class groups

Let S be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of S is finite and at least 4, then the isomorphism type of the pure mapping class group associated to S, denoted PMap(S), detects the homeomorphism type of S. As a corollary, every automorphism of PMap(S) is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that PMap(S) is residually finite if and only if S has finite genus, demonstrating that the algebraic structure of PMap(S) can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that Map(S) fails to be residually finite for any infinite-type surface S. In addition, we give a topological generating set for PMap(S) equipped with the compact-open topology. In particular, if S has at most one end accumulated by genus, then PMap(S) is topologically generated by Dehn twists, otherwise the Dehn twists along with handle shifts topologically generate.