Counting surface subgroups in cusped hyperbolic 3-manifolds

Abstract

Let $M={\mathbb{H}}^3/\hat{I}“$ be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of $\hat{I}“$ (up to conjugacy and commensurability) of genus at most $g$ is bounded both above and below by functions of the form $(cg{)}^{2g}$. As a corollary, for all $h\ge 4$, the number of purely pseudo-Anosov closed surface subgroups of genus at most $g$ of the mapping class group $\mathrm{Mod}(S_{h,0})$ is bounded below by $(Cg{)}^{2g}$ for a universal constant $C$. In contrast, for some $g\ge 2$, we construct infinitely many conjugacy classes of genus-$g$ surface subgroups of $\hat{I}“$ with accidental parabolics.

25 pages, 3 figures