[Paper Review] Non-Abelian number theory and the structure of curves on surfaces

Moira Chas|arXiv (Cornell University)|Aug 9, 2016

Mathematical Dynamics and Fractals4 references3 citations

TL;DR

This paper investigates the combinatorics of curves on the torus with one boundary component by analyzing orbits under the mapping class group, revealing that the number of curves of fixed self-intersection number and word length grows asymptotically as a rational multiple of $ \ell^2 $, with coefficients tied to the Euler totient function. The key contribution is a novel connection between non-Abelian number theory, hyperbolic geometry, and arithmetic via Mirzakhani’s asymptotics, leading to six precise conjectures on the rationality and structure of growth coefficients.

ABSTRACT

In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a certain length, either geometric or combinatorial. Some cases are rigurolosly determined and the Euler totient function emerges. More complicated orbits are computed to suggest an array of formulae continuing to involve the Euler totient function. We formulate six precise conjectures. These include a novel study of an "inverse" function of the Mirzakhani's asymptotics. The geometric part of our study was motivated by the rationality aspect of these asymptotics.

Motivation & Objective

• To understand the growth of curves on the torus with one boundary component under the action of the mapping class group, particularly those with fixed self-intersection number.
• To overcome computational challenges from exponential growth in curve enumeration by focusing on orbits of the mapping class group up to a given length.
• To establish a connection between combinatorial word length growth and geometric length asymptotics, especially through the lens of Mirzakhani’s work.
• To formulate and test conjectures linking the Euler totient function, rational coefficients, and the structure of curve orbits on surfaces.
• To extend the observed patterns to general hyperbolic surfaces of genus $ g $ and $ n $ boundary components, proposing a framework for non-Abelian number theory.

Proposed method

• Count unoriented curves in orbits of the mapping class group $ M(1,1) $ up to a given word length $ \ell $, using cyclically reduced words in the fundamental group generators $ a, b, A, B $.
• Use the Euler totient function $ \Phi(n) $ to compute exact counts for orbits with self-intersection number $ k = 0,1,2,3 $, showing that $ c_\ell(\alpha) = 2\sum \Phi(\cdot) $.
• Construct explicit hyperbolic metrics on the one-holed torus to compute geometric lengths of curves, enabling comparison with word-length-based asymptotics.
• Compare Mirzakhani’s geometric length asymptotics with the combinatorial word-length results, finding agreement in rational coefficients.
• Use numerical experiments and graphical analysis to support conjectures on the asymptotic behavior of curve counts, including inverse functions of Mirzakhani’s growth.
• Generalize observed patterns to higher genus and boundary components, proposing a non-Abelian number theory framework rooted in arithmetic and topology.

Experimental results

Research questions

• RQ1 How does the number of curves of fixed self-intersection number and word length grow in the orbit of a given curve on the one-holed torus?
• RQ2 Can the growth coefficients of such orbits be expressed rationally in terms of the Euler totient function, and do they match Mirzakhani’s geometric asymptotics?
• RQ3 Is there a functional relationship between the inverse of Mirzakhani’s asymptotic function and the geometric length of curves in a given orbit?
• RQ4 Do the coefficients of the $ \ell^2 $-asymptotic growth in word length match rational constants derived from geometric length counts?
• RQ5 Can the observed patterns of growth, involving the Euler totient function and rational coefficients, be generalized to all hyperbolic surfaces of genus $ g $ and $ n $ boundary components?

Key findings

• For self-intersection number $ k = 0 $, the number of curves of word length $ \ell $ in the orbit of $ \alpha $ is exactly $ 2\Phi(\ell) $, and the cumulative count up to $ \ell $ is asymptotic to $ \frac{6}{\pi^2}\ell^2 $.
• For $ k = 1 $ and partially for $ k = 2 $, the number of curves of word length $ \ell $ in each orbit is exactly expressible as a sum of Euler totient functions, yielding rational growth coefficients.
• The geometric length asymptotics of curves in each orbit match the word-length-based coefficients, with Mirzakhani’s constant factoring into topology, geometry, and orbit-specific parts, all rational.
• The sum of growth coefficients $ p_\alpha $ over all orbits of curves with self-intersection number $ k $ is within one of the total number of such orbits, suggesting deep arithmetic structure.
• Conjecture 4 posits that all such curve counts are of the form $ 2\sum \Phi\left(\frac{\ell + j_i}{k_i}\right) $, implying asymptotic growth $ \frac{6}{\pi^2} p \ell^2 $ with rational $ p $, and this is supported by data for $ k \leq 3 $.
• The observed rationality of growth coefficients in both word and geometric length settings suggests a broader non-Abelian number theory framework for hyperbolic surfaces.