[Paper Review] Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold

Jeremy Kahn, Vladimir Marković|arXiv (Cornell University)|Dec 13, 2010

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper establishes the asymptotic growth rate of the number of conjugacy classes of surface subgroups of genus $ g $ in the fundamental group of a closed hyperbolic 3-manifold. Using triangulation counting and maximal surface group enumeration, it proves that this number grows as $ (c_1 g)^{2g} $ to $ (c_2 g)^{2g} $, with precise bounds depending on the injectivity radius, resolving a key question in 3-manifold topology and geometric group theory.

ABSTRACT

Let M be a closed hyperbolic 3-manifold. We show that the number of genus g surface subgroups of the fundamental group of M grows like g^{2g}.

Motivation & Objective

• To determine the asymptotic growth rate of the number of conjugacy classes of surface subgroups of genus $ g $ in $ \pi_1(M^3) $ for a closed hyperbolic 3-manifold $ M^3 $.
• To improve upon prior bounds by Masters, providing a tight $ (c g)^{2g} $ estimate for both conjugacy classes and commensurability classes of surface subgroups.
• To establish an unconditional lower bound using maximal surface group enumeration, replacing conditional assumptions in earlier work.
• To unify geometric and group-theoretic techniques—specifically triangulations and covering space theory—to count surface subgroups precisely.

Proposed method

• Construct a triangulation $ \tau \in \mathcal{T}(k,g) $ on a genus $ g $ surface with bounded vertex degree and edge count, using a Delaunay triangulation of a maximal $ s/4 $-ball packing in an $ s $-thick Riemann surface.
• Use the injectivity radius $ s $ of $ M^3 $ to ensure that any pleated surface in $ M^3 $ lifts to a thick surface admitting such a triangulation.
• Count the number of genus $ g $ surface subgroups by counting suitable triangulations and their lifts to $ M^3 $, bounding the number via combinatorial enumeration of graphs with $ O(g) $ vertices and edges.
• Apply results from Muller and Puchta on maximal surface group enumeration in surface groups to obtain a lower bound on the number of maximal covers of genus $ g_0 $ surfaces.
• Construct new surface subgroups by amalgamating two $ n $-fold covers along $ k $-degree lifts of non-separating curves, ensuring maximality and injectivity.
• Use the formula $ g_n = n(2g_0 - 1) $ to relate the genus of the resulting surface to the degree of the cover, enabling asymptotic counting.

Experimental results

Research questions

• RQ1 How does the number of conjugacy classes of surface subgroups of genus $ g $ grow in the fundamental group of a closed hyperbolic 3-manifold?
• RQ2 Can the upper bound on the number of such surface subgroups be improved from $ g^{c g} $ to $ (c g)^{2g} $, and what combinatorial structure enables this?
• RQ3 What is the minimal growth rate of the number of commensurability classes of surface subgroups, and can it be bounded unconditionally?
• RQ4 How does the structure of maximal surface subgroups in surface groups contribute to constructing new surface subgroups in $ \pi_1(M^3) $?
• RQ5 What is the precise asymptotic order of $ \log s_i(M^3, g) / (2g \log g) $ as $ g \to \infty $?

Key findings

• The number of conjugacy classes of surface subgroups of genus at most $ g $, denoted $ s_2(M^3, g) $, satisfies $ s_2(M^3, g) \leq (c_2 g)^{2g} $ for some $ c_2 > 0 $ depending only on the injectivity radius of $ M^3 $.
• The number of commensurability classes of such surface subgroups, $ s_1(M^3, g) $, satisfies $ (c_1 g)^{2g} \leq s_1(M^3, g) $ for some $ c_1 > 0 $, with the bound being unconditional.
• The asymptotic growth of both $ s_1(M^3, g) $ and $ s_2(M^3, g) $ is precisely $ (c g)^{2g} $, as confirmed by the limit $ \lim_{g \to \infty} \frac{\log s_i(M^3, g)}{2g \log g} = 1 $ for $ i = 1,2 $.
• The lower bound is established via maximal surface group enumeration: for large $ n $, the number of maximal $ n $-fold covers of a genus $ g_0 $ surface is $ (n!)^{g_0 - 2}(1 + o(1)) $.
• For some $ 1 \leq k \leq n-1 $, the number of maximal $ n $-fold covers where a fixed non-separating curve lifts with degree $ k $ exceeds $ ((n-1)!)^{g_0 - 2}(1 + o(1)) $.
• The construction of new surface subgroups by amalgamating two $ n $-fold covers along $ k $-degree lifts of non-separating curves yields maximal surface subgroups in $ \pi_1(M^3) $, enabling the lower bound.