Mapping class groups of covers with boundary and braid group embeddings

TL;DR

This paper classifies when liftable and symmetric mapping class groups of surface covers coincide with the full group, and constructs new non-geometric braid group embeddings into mapping class groups using Dehn twists.

Contribution

It provides a classification of when certain subgroups of mapping class groups equal the entire group and introduces novel non-geometric braid embeddings via Dehn twists.

Findings

Classification of when liftable and symmetric mapping class groups equal the full group

Construction of infinite families of non-geometric braid embeddings

Use of Birman-Hilden theorem and fundamental groupoid actions

Abstract

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.