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

TL;DR

This paper explores the monodromy of strata of abelian differentials, showing the fundamental group surjects onto framed mapping class groups, which are finitely generated, providing a detailed understanding of their structure.

Contribution

It establishes a complete characterization of the monodromy group for strata of abelian differentials and proves that framed mapping class groups are finitely generated with explicit generators.

Findings

Fundamental group surjects onto framed mapping class groups.

Framed mapping class groups are finitely generated.

Explicit generating sets for these groups are provided.

Abstract

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.