Big pure mapping class groups are never perfect

Abstract

We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to ⊕ 2 ℵ 0 Q. We also extend this to the Torelli group and show that in the case of surfaces with infinite genus the abelianization of the Torelli group contains an indivisible copy of ⊕ 2 ℵ 0 Z as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to Q.

Big mapping class groups and the co-Hopfian property

“…is exact; see [10]. The infinite type case is proved by Dickmann-Domat in the appendix to Domat's paper [13].…”

“…Thanks also to AIM for its hospitality and financial support. We would like to thank Nick Vlamis and Henry Wilton for enlightening conversations about Lemma 6.2, and Vlamis for the reference [13].…”

Mapping class groups with the Rokhlin property

“…Conner's conjecture would suggest that to find a normal subgroup of countably infinite index, then one would need to construct a homomorphism to the rationals: in [7], Domat and Dickmann do just that for the case of the mapping class group of the Loch Ness monster surface. Their homomorphism is (necessarily) discontinuous; in contrast, in the case of the mapping class group of the Cantor tree surface, where every homomorphism to the rationals must be continuous [21] (see discussion below on automatic continuity), the second author [33] has shown that no normal countable-index subgroups exist.…”

“…Note, this result does not preclude a mapping class group from having the ACP; it only prevents the approach using ample generics. In fact, the question of which mapping class groups have the ACP has turned out to be quite complicated: it is known that the mapping class group of the 2-sphere minus a Cantor set has the ACP [21] and the mapping class group of the Loch Ness monster does not [7]. (Mann [21] gives other examples of mapping class groups with and without the ACP.…”

Three perfect mapping class groups

“…The motivation of that conjecture was a result of Patel and the author [17] showing that in those cases the pure mapping class group had a dense perfect subgroup (the perfectness follows from the fact that pure mapping class groups of finite-type surfaces of genus at least three are perfect, see [10,Theorem 5.2]). Domat (with an appendix with Dickmann, which is relevant in the case of the Loch Ness monster below) [8] disproved this conjecture by showing the existence of homomorphisms from pure mapping class groups to the rationals whenever the underlying surface is of infinite type and has at most one nonplanar end. Now, there are two infinite-type surfaces without boundary in which the mapping class group is equal to the pure mapping class group, namely the Loch Ness monster surface-that is, the borderless one-ended infinite-genus surface-and the surface obtained from removing a single point from the Loch Ness monster surface.…”