Framed mapping class groups and strata of Abelian differentials

(joint with Nick Salter)

Framed mapping class groups and the monodromy of strata of Abelian differentials

Link to most recent version (1/31/20)

This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. Building off of prior work of the authors, we show that the fundamental group of a stratum surjects onto the subgroup of the mapping class group which preserves a fixed framing of the underlying Riemann surface, thereby giving a complete characterization of the monodromy group. In the course of our proof we also show that these “framed mapping class groups” are finitely generated (even though they are of infinite index) and give explicit generating sets.

Relative homological representations of framed mapping class groups

Link to most recent version (2/6/20)

Let (Σg,Z) be a surface endowed with a nonempty finite set of marked points. When Σg\Z is equipped with a preferred framing φ, the “framed mapping class group” PMod(Σg,Z)[φ] is defined as the subgroup of the (pure) mapping class group PMod(Σg,Z) consisting of elements that preserve φ up to isotopy. Such groups arise naturally in the study of families of translation surfaces. In this note, we determine the action of PMod(Σg,Z)[φ] on the relative homology H1g,Z;Z), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.