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A Moebius inversion formula to discard tangled hyperbolic surfaces

Nalini Anantharaman, Laura Monk

TL;DR

This work introduces a Moebius inversion framework to sieve out tangles from random hyperbolic surfaces, enabling systematic conditioning on tangle-free geometries. Building on Friedman's ideas, it defines a big moduli space and a derived-tangle calculus that yields a unique, multiplicative Möbius function μ supported on tangled configurations, with the key identity 1 = sum μ(τ) over sub-surfaces. The authors then prove a polynomial bound for the number of local topological types of short geodesics in tangle-free surfaces, obviating the exponential proliferation seen in tangled cases. Together, these results provide a robust method to study spectral and geometric statistics under tangle-free conditioning and offer a general framework potentially applicable to other geometric pattern constraints in moduli spaces.

Abstract

Recent literature on Weil-Petersson random hyperbolic surfaces has met a consistent obstacle: the necessity to condition the model, prohibiting certain rare geometric patterns (which we call tangles), such as short closed geodesics or embedded surfaces of short boundary length. The main result of this article is a Moebius inversion formula, allowing to integrate the indicator function of the set of tangle-free surfaces in a systematic, tractable way. It is inspired by a key step of Friedman's celebrated proof of Alon's conjecture. We further prove that our tangle-free hypothesis significantly reduces the number of local topological types of short geodesics, replacing the exponential proliferation observed on tangled surfaces by a polynomial growth.

A Moebius inversion formula to discard tangled hyperbolic surfaces

TL;DR

This work introduces a Moebius inversion framework to sieve out tangles from random hyperbolic surfaces, enabling systematic conditioning on tangle-free geometries. Building on Friedman's ideas, it defines a big moduli space and a derived-tangle calculus that yields a unique, multiplicative Möbius function μ supported on tangled configurations, with the key identity 1 = sum μ(τ) over sub-surfaces. The authors then prove a polynomial bound for the number of local topological types of short geodesics in tangle-free surfaces, obviating the exponential proliferation seen in tangled cases. Together, these results provide a robust method to study spectral and geometric statistics under tangle-free conditioning and offer a general framework potentially applicable to other geometric pattern constraints in moduli spaces.

Abstract

Recent literature on Weil-Petersson random hyperbolic surfaces has met a consistent obstacle: the necessity to condition the model, prohibiting certain rare geometric patterns (which we call tangles), such as short closed geodesics or embedded surfaces of short boundary length. The main result of this article is a Moebius inversion formula, allowing to integrate the indicator function of the set of tangle-free surfaces in a systematic, tractable way. It is inspired by a key step of Friedman's celebrated proof of Alon's conjecture. We further prove that our tangle-free hypothesis significantly reduces the number of local topological types of short geodesics, replacing the exponential proliferation observed on tangled surfaces by a polynomial growth.
Paper Structure (32 sections, 23 theorems, 76 equations, 1 figure)

This paper contains 32 sections, 23 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. History of the word tangle in the graph setting
  3. Tangles in hyperbolic surfaces
  4. Polynomial counting result
  5. Conditioning on the set of tangle-free surfaces
  6. Moebius inversion formulas in number theory and for hyperbolic surfaces
  7. Our Moebius inversion formula
  8. Tangles in hyperbolic surfaces
  9. The big moduli space $\mathbf{M}$
  10. c-surfaces
  11. Extended notion of moduli space
  12. Sub-c-surfaces and their geodesic representatives
  13. Subset filling a c-surface
  14. Definition of tangles
  15. Derived tangles
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\mathrm{Loc}_{\chi}^{\kappa,R, L}$ be the set of local topological types of closed geodesics of length $\leq L$ that can arise in a $(\kappa, R)$-tangle-free surface of Euler characteristic $-\chi$. Then, where $\mathcal{O}_{ \chi} \left( \cdot \right)$ is the usual Landau "big-O" notation, with an implied constant depending on $\chi$.

Figures (1)

  • Figure 1: Four cases for the volute $G(x_0)$ and the expression for $h$ in each case.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['t:moebius']}
  • Corollary 1.3
  • Definition 2.2
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • ...and 47 more