Counting immersed surfaces in hyperbolic 3–manifolds

Abstract

We count the number of conjugacy classes of maximal, genus $g$, surface subroups in hyperbolic 3–manifold groups. For any closed hyperbolic 3–manifold, we show that there is an upper bound on this number which grows factorially with $g$. We also give a class of closed hyperbolic 3–manifolds for which there is a lower bound of the same type.