Nielsen realization for infinite-type surfaces

Rylee Lyman (Rutgers)

Abstract: Recently the theory of infinite-type surfaces—those whose fundamental group is not finitely generated—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.