Allcock, building on the work of Greenburg, proved that for any countable group G, there is a complete hyperbolic surface whose isometry group is exactly G. When the group is finite, Allcock’s construction yields a closed surface, but when it is not finite, the construction gives an infinite-genus surface. Moreover, the topology of the surface constructed depends on the choice of group G.
In this talk, we discuss a related question. We fix any infinite-genus surface S and characterize all groups that can arise as the isometry group for a complete hyperbolic structure on S. We will then discuss how to leverage these results to give algebraic invariants of the homeomorphism, diffeomorphism, and mapping class groups of S. This talk is based on joint work with T. Aougab and N. Vlamis.