Big Torelli groups: generation and commensuration

Abstract

For any surface Σ of infinite topological type, we study the Torelli subgroup I(Σ) of the mapping class group MCG(Σ), whose elements are those mapping classes that act trivially on the homology of Σ. Our first result asserts that I(Σ) is topologically generated by the subgroup of MCG(Σ) consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman [4], Powell [24], and Putman [25] we deduce that I(Σ) is topologically generated by separating twists and bounding pair maps. Next, we prove the abstract commensurator group of I(Σ) coincides with MCG(Σ). This extends the results for finite-type surfaces [9,6,7,16] to the setting of infinite-type surfaces.

Big mapping class groups: an overview

“…Adapting an argument presented in [45,Theorem 7.16] showing that the mapping class group of a finite-type surface is generated by torsion elements, Afton-Freedman-Lanier-Yin [1] observed: Combining results of Birman [24], Powell [98] and an argument due to Justin Malestein, the above theorem implies the following (see [10] for details and definitions): Theorem 4.7 ( [10]). Let S be any surface of infinite type.…”

“…In [10], it shown that I(S) is also algebraically rigid; more concretely: The equivalent statement for finite-type surfaces was proved by Farb-Ivanov [44] for automorphisms, and by Brendle-Margalit [27] for commensurations.…”

Big pure mapping class groups are never perfect

“…In [3] the authors found a topological generating set for I(S) when S is infinite type. Theorem 9.2 ([3], Corollary 2).…”

Big Mapping Class Groups With Hyperbolic Actions: Classification and Applications

“…Further developments, providing actions of Map(Σ) on hyperbolic graphs under various topological conditions on the surface, include [3,20,21], for instance. The study of the geometry of such graphs is still in constant expansion (see, e.g., [4,7,26]).…”