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On trace set of hyperbolic surfaces and a conjecture of Sarnak and Schmutz

Yanlong Hao

TL;DR

This work studies the trace set $Tr(\Gamma)$ of Fuchsian lattices and its arithmetic implications. It proves a strong version of Schmutz’s conjecture for non-uniform lattices: linear growth of $Tr(\Gamma)$ characterizes arithmeticity, and demonstrates that for a fixed genus $g\ge 3$ the cocompact embeddings with trace-growth exceeding $n^{2-\varepsilon}$ occupy positive Weil-Petersson volume, with an explicit asymptotic description. The authors adapt Schmutz’s geometric ideas into a powerful algebraic framework, establishing rationality properties of derived subgroups and employing density arguments via $\mathbb{Z}$-affine constructions and Dirichlet-type estimates to constrain trace sets. They extend trace-set growth analysis to the cocompact regime on high-genus surfaces, using multicurves, Cheeger constants, and Mirzakhani’s growth results to show WP-volume prevalence of large-growth instances. Overall, the paper links trace-set growth to arithmeticity and provides volume-analytic insights into trace-growth phenomena on moduli spaces of hyperbolic surfaces.

Abstract

In this paper, we investigate the trace set of a Fuchsian lattice. There are two results of this paper: the first is that for a non-uniform lattice, we prove Scmutz's conjecture: the trace set of a Fuchsian lattice exhibits linear growth if and only if the lattice is arithmetic. Additionally, we show that for a fixed surface group of genus bigger than 2 and any positive number $ε$, te set of cocompact lattice embedding such that their growth rate of trace set exceeds $n^{2-ε}$ has positive Weil-Petersson volume. We also provide an asymptotic analysis of the volume of this set.

On trace set of hyperbolic surfaces and a conjecture of Sarnak and Schmutz

TL;DR

This work studies the trace set of Fuchsian lattices and its arithmetic implications. It proves a strong version of Schmutz’s conjecture for non-uniform lattices: linear growth of characterizes arithmeticity, and demonstrates that for a fixed genus the cocompact embeddings with trace-growth exceeding occupy positive Weil-Petersson volume, with an explicit asymptotic description. The authors adapt Schmutz’s geometric ideas into a powerful algebraic framework, establishing rationality properties of derived subgroups and employing density arguments via -affine constructions and Dirichlet-type estimates to constrain trace sets. They extend trace-set growth analysis to the cocompact regime on high-genus surfaces, using multicurves, Cheeger constants, and Mirzakhani’s growth results to show WP-volume prevalence of large-growth instances. Overall, the paper links trace-set growth to arithmeticity and provides volume-analytic insights into trace-growth phenomena on moduli spaces of hyperbolic surfaces.

Abstract

In this paper, we investigate the trace set of a Fuchsian lattice. There are two results of this paper: the first is that for a non-uniform lattice, we prove Scmutz's conjecture: the trace set of a Fuchsian lattice exhibits linear growth if and only if the lattice is arithmetic. Additionally, we show that for a fixed surface group of genus bigger than 2 and any positive number , te set of cocompact lattice embedding such that their growth rate of trace set exceeds has positive Weil-Petersson volume. We also provide an asymptotic analysis of the volume of this set.
Paper Structure (22 sections, 22 theorems, 122 equations)

This paper contains 22 sections, 22 theorems, 122 equations.

Table of Contents

  1. Introduction and Statement of the main results
  2. Related works
  3. Further research and open questions
  4. Definitions, notations, and some preliminaries
  5. Fuchsian groups
  6. Characterization of arithmetic Fuchsian and Kleinian groups
  7. $\mathbb{Z}$-GCD and $\mathbb{Z}$-LCM of real numbers
  8. Natural density of $\mathbb{Z}$-affine space
  9. Dirichlet's Theorem on arithmetic progressions
  10. Critical exponent and Cheeger constant
  11. second order linear recurrence sequences
  12. Proof of Theorem \ref{['main2']}
  13. Case 1: $\Gamma$ is elementary
  14. Case 2: $\Gamma$ is non-elementary
  15. First step: $\Gamma^{(2)}$ is rational.
  16. ...and 7 more sections

Key Result

Theorem A

Let $\Gamma$ be a non-uniform lattice of $\operatorname{PSL}(2,\mathbb{R})$. If then $\Gamma$ is arithmetic.

Theorems & Definitions (42)

  • Conjecture 1.1: Sarnak MR1321639
  • Conjecture 1.2: Schmutz MR1394753
  • Theorem A
  • Theorem 1.3
  • Theorem B
  • Example 1.4
  • Corollary 1.5
  • Theorem C
  • Remark 1.6
  • Theorem D
  • ...and 32 more