Algebraic and topological properties of big mapping class groups

Abstract

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.

Big mapping class groups and rigidity of the simple circle

“…It is the best understood and most studied example of a so-called 'big' mapping class group; see e.g. [1,10,15].…”

Isomorphisms Between Big Mapping Class Groups

“…Unlike mapping class groups of finite-type surfaces, these big mapping class groups have uncountably many elements and inherit a non discrete topology from the compact open topology on Homeo(S). Despite a recent growing interest in big mapping class groups (e.g., [Cal,AFP,DFV,PV,HMV1]), the above properties have remained open in this setting. Our main results establish them for all infinite-type surfaces.…”

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

“…Loop shifts are the graph equivalent of handle shifts on surfaces, which were introduced by Patel and Vlamis in [13]. Let Λ be the graph in standard form with exactly two ends, each of which are accumulated by loops, as in Figure 7.…”