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Counting surface subgroups in cusped hyperbolic 3-manifolds

Xiaolong Hans Han, Zhenghao Rao, Jia Wan

Abstract

Let $M =\mathbb{H}^3/Γ$ be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of $Γ$ (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 $Γ$ with accidental parabolics.

Counting surface subgroups in cusped hyperbolic 3-manifolds

Abstract

Let be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of (up to conjugacy and commensurability) of genus at most is bounded both above and below by functions of the form . As a corollary, for all , the number of purely pseudo-Anosov closed surface subgroups of genus at most of the mapping class group is bounded below by for a universal constant . In contrast, for some , we construct infinitely many conjugacy classes of genus- surface subgroups of with accidental parabolics.
Paper Structure (11 sections, 20 theorems, 27 equations, 3 figures)

This paper contains 11 sections, 20 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Upper Bound
  3. The Lower Bound
  4. Review of Kahn-Wright's Construction
  5. Pants and Hamster Wheels
  6. Well-matched Components and QF Surfaces
  7. Good QF Representations and Bending Deformations
  8. Assemble good representations
  9. Counting Maximal QF Surface Subgroups
  10. Counting Purely Pseudo-Anosov Surface Subgroups
  11. Coannular Surfaces

Key Result

Theorem 1.1

Let $M=\mathbb{H}^3/\Gamma$ be a cusped hyperbolic 3-manifold. Then there exist positive constants $C_1,C_2$, depending only on $M$, such that for $g$ sufficiently large, In fact, the constant $C_1$ depends only on the systole and the volume of a compact core of $M$.

Figures (3)

  • Figure 1: A hamster wheel.
  • Figure 2: A possible compressing disk in $M_F$.
  • Figure 3: The gluing construction for the case of $m=2$, where $F_1$ and $F_2$ coincide in $M$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 32 more