Seminar

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Building on the work of W.Thurston on postcritically finite maps, A.Epstein defined deformation spaces of holomorphic maps. Epstein used deformation techniques to obtain transversality results for varieties defined by critical relations. We study further topological properties of deformation spaces. We will discuss theorems and

conjectures on the structure of the boundary of deformation spaces. We introduce equalizing multicurves (an analog of W.Thurston's invariant multicurves) and connect them to the boundary behavior and to the subgroups of the mapping class groups preserving deformation spaces. Our main examples are Per_n curves, critically periodic quadratic rational maps, and their deformation spaces. In particular, we will show that these deformation spaces are not contractible, answering a question by A. Epstein. This is joint work with J.Kahn and N.Selinger.