A perspective projection of a dodecahedral tessellation in H3. Four dodecahedra meet at each edge, and eight meet at each vertex, like the cubes of a cubic tessellation in E3.

Title: Do You Even Lift – Branched Covers and Liftable Mapping Classes

Speaker: Ananya Satoskar

Abstract:The mapping class group is the group of all orientation-preserving self-homeomorphisms of a manifold (upto isotopy). Determining the algebraic structure of mapping class groups of two-manifolds is of great importance because of their connections to the theory of Riemann surfaces. One way to study this is by investigating group homomorphisms between mapping class groups – a folkloric conjecture by Mirzakhani informally states that every such homomorphism is induced by some geometric manipulation of the associated surfaces. Classical work by Birman and Hilden established a way to relate certain finite-index subgroups of mapping class groups using the topological structure of an underlying covering space. I will define mapping class groups, review recent work in this direction and introduce a more modern combinatorial approach to Birman-Hilden theory that lets us investigate many questions about irregular branched covers and the lifting of maps.

Some snacks will be provided before and after the talk.

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