Counting surface subgroups in cusped hyperbolic 3-manifolds

TL;DR

This paper investigates the enumeration of surface subgroups within cusped hyperbolic 3-manifolds, establishing bounds on their quantity and constructing examples with specific properties, advancing understanding of their distribution and complexity.

Contribution

It provides bounds on the number of surface subgroups of hyperbolic 3-manifolds and constructs examples with accidental parabolics, offering new insights into their structure.

Findings

Number of quasi-Fuchsian surface subgroups grows roughly as (cg)^{2g}

Lower bounds on the count of pseudo-Anosov surface subgroups in mapping class groups

Existence of infinitely many surface subgroups with accidental parabolics

Abstract

Let $M =\mathbb{H}^3/\Gamma$ be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of $\Gamma$ (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 $h\geq 4$, 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 $g \geq 2$, we construct infinitely many conjugacy classes of genus-$g$ surface subgroups of $\Gamma$ with accidental parabolics.