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Homomorphisms between mapping class groups of surfaces

Banff International Research Station for Mathematical Innovation and Discovery

A general problem is to understand all (injective) homomorphisms between (finite index subgroups of) mapping class groups of surfaces. Birman and Hilden proved that if $S\rightarrow X$ is a regular branched covering space of surfaces, there is an embedding of the subgroup of the mapping class group of $X$ consisting of mapping classes that have representatives that lift to $S$ in the mapping class group of $S$ modulo the group of deck transformations. This relationship does not always hold for irregular branched covers. We give a necessary condition and a sufficient condition for when such an embedding exists. We also give new explicit examples that satisfy the necessary condition and examples that do not satisfy the sufficient condition.

Mathematics; Manifolds and cell complexes; Group theory and generalizations; Discrete mathematics

video/mp4

Author affiliation: University of Wisconsin-Milwaukee

2018-04-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65379

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/