Mapping class groups of covers with boundary and braid group embeddings

Ghaswala, Tyrone; McLeay, Alan

doi:10.2140/agt.2020.20.239

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Mapping class groups of covers with boundary and braid group embeddings

Tyrone Ghaswala and Alan McLeay

Algebraic & Geometric Topology 20 (2020) 239–278

DOI: 10.2140/agt.2020.20.239

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 nongeometric 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.

Keywords

mapping class groups, braid groups, covering spaces

Mathematical Subject Classification 2010

Primary: 57M12

Secondary: 20F36, 20L05

References

Publication

Received: 11 May 2018

Revised: 11 March 2019

Accepted: 1 April 2019

Published: 23 February 2020

Authors

Tyrone Ghaswala
Department of Mathematics University of Manitoba Winnipeg, MB Canada

Alan McLeay
Mathematics Research Unit University of Luxembourg Esch-sur-Alzette Luxembourg

Volume 20, issue 1 (2020)