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.

It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

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Graphs of curves on infinite-type surfaces with mapping class group actions

It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.