三个完美映射类群

Big pure mapping class groups are never perfect

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 an uncountable direct sum of rationals. 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 an uncountable free abelian group as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to the rationals.

具有非紧边界曲面的映射类群

Mapping class groups of surfaces with noncompact boundary components

本文证明了一类具有非紧边界的无限型曲面的纯映射类群一致完全，结合近期工作给出了无限型曲面纯映射类群完全与一致完全的完整分类，并开发了将一般曲面切割为简单曲面的推广方法。

Bounded cohomology and non-uniform perfection of mapping class groups

Using the existence of certain symplectic submanifolds in symplectic 4-manifolds, we prove an estimate from above for the number of singular fibers with separating vanishing cycles in minimal Lefschetz fibrations over surfaces of positive genus. This estimate is then used to deduce that mapping class groups are not uniformly perfect, and that the map from their second bounded cohomology to ordinary cohomology is not injective.

Self-similar surfaces: involutions and perfection

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the ot…

纯映射类群的第一整上同调

The first integral cohomology of pure mapping class groups

该研究突破Powell关于有限型高亏格曲面纯映射类群的经典完满性结论，构造无穷亏格映射类群到整数的非平凡同态，计算亏格≥2连通可定向曲面纯映射类群的第一整上同调，并证明无穷亏格情形下该群可分解为半直积。

非平面端点始终是连续不可忘的

Non-planar ends are continuously unforgettable

该研究证明了特定映射类群子群之间的连续满射由同胚诱导，并推导出这些群是霍普夫拓扑群的结论。