Mapping class groups of covers with boundary and braid group embeddings

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.

Finite presentations for the balanced superelliptic mapping class groups

“…Since f [ ϕ] −1 lies in the kernel of θ, there exists 0 By Lemma 4.6 and Proposition 4.7, we regard SMod g, * ;k as a subgroup of SMod g;k which consists of elements represented by symmetric elements preserving p 2n+2 . The next lemma is given by Proposition 2.1 in [7].…”

“…Mod g, * and Mod 1 g ) which consists of elements represented by symmetric homeomorphisms. Ghaswala and McLeay [7] showed that any element in SMod 1 g;k are represented by a homeomorphism which commutes with ζ. Birman and Hilden [4] showed that SMod g;k (resp. SMod g, * ;k and SMod 1 g;k ) coincides with the group of symmetric isotopy classes of symmetric homeomorphisms on Σ g (resp.…”

“…Acknowledgement: The authors would like to express their gratitude to Tyrone Ghaswala for helpful advices and telling them Proposition 5.3 in [7] that is a generalized version of Lemma 6.1. The first author was supported by JSPS KAKENHI Grant Numbers JP16K05156 and JP20K03618.…”

“…a k ∪ a 1 ). Under the situation above, we have the following lemma that is a specialized version of Proposition 5.3 in [7]. 21).…”

From braid groups to mapping class groups

“…The monodromy of the degree d super-hyperelliptic family has been computed by [McM13] and [GM20]. By the Hurwitz formula, the genus of this family is (d − 1)(k − 1) for n = 2k.…”