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Nielsen realization for infinite-type surfaces

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Rylee Lyman (Rutgers)

Abstract: Recently the theory of infinite-type surfacesâ€”those whose
fundamental group is not finitely generatedâ€”has generated a lot of
interest among geometric group theorists and low-dimensional
topologists. Mapping class groups of infinite-type surfaces share some
features with their finite-type cousins but are in many ways much
wilder. In 1983 Kerckhoff solved the "Nielsen realization" problem posed
in 1932: finite subgroups of the mapping class group of a finite-type
surfaces of negative Euler characteristic are exactly the groups of
isometries of some hyperbolic metric on the surface. Recently, joint
with Santana Afton, Danny Calegari and Lvzhou Chen, I extended this
theorem to orientable, infinite-type surfaces. I'd like to introduce
infinite-type surfaces and discuss the theorem and some of its
consequences.