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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

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.

mapping class groups, braid groups, covering spaces
Mathematical Subject Classification 2010
Primary: 57M12
Secondary: 20F36, 20L05
Received: 11 May 2018
Revised: 11 March 2019
Accepted: 1 April 2019
Published: 23 February 2020
Tyrone Ghaswala
Department of Mathematics
University of Manitoba
Winnipeg, MB
Alan McLeay
Mathematics Research Unit
University of Luxembourg