The mapping class group of a surface is the group of isotopy classes of boundary preserving homeomorphisms of the surface. Given a finite sheeted covering space between surfaces, we may ask what relationship, if any, exists between the two mapping class groups?

In joint work with Tyrone Ghaswala we investigate this question for surfaces with non-empty boundary.  I will discuss a classical theorem of Birman-Hilden and give new insight into a family of covering spaces related to the Burau representation of braid groups.

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