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A new proof of Chen's theorem for Markoff graphs

Daniel E. Martin

TL;DR

The paper addresses the problem of Chen's theorem, which implies that every connected component of the Markoff graph modulo a prime $p$ has size divisible by $p$, supporting Baragar's conjecture for almost all primes. It provides a self-contained second proof by translating Markoff solutions to Penner coordinates with $y_1+y_2+y_3=1$ and establishing edge-wise identities $y_1+y_1'=1$, leading to a global divisibility conclusion. The approach yields a clear combinatorial and coordinate-based argument that aligns with the framework of previous results by Bourgain–Gamburd–Sarnak. This strengthens the understanding of Markoff graphs over finite fields and reinforces Baragar's conjecture up to finitely many exceptional primes.

Abstract

In 2021, Chen proved a congruence for the degree of a certain map on the space of covers of elliptic curves. He concluded as a corollary that the size of any connected component of the Markoff mod $p$ graph is divisible by $p$. In combination with the work of Bourgain, Gamburd, and Sarnak, Chen's result proves a conjecture of Baragar for all but finitely many primes: the Markoff mod $p$ graph is connected. In this note, we provide an alternative proof for the Markoff corollary of Chen's theorem.

A new proof of Chen's theorem for Markoff graphs

TL;DR

The paper addresses the problem of Chen's theorem, which implies that every connected component of the Markoff graph modulo a prime has size divisible by , supporting Baragar's conjecture for almost all primes. It provides a self-contained second proof by translating Markoff solutions to Penner coordinates with and establishing edge-wise identities , leading to a global divisibility conclusion. The approach yields a clear combinatorial and coordinate-based argument that aligns with the framework of previous results by Bourgain–Gamburd–Sarnak. This strengthens the understanding of Markoff graphs over finite fields and reinforces Baragar's conjecture up to finitely many exceptional primes.

Abstract

In 2021, Chen proved a congruence for the degree of a certain map on the space of covers of elliptic curves. He concluded as a corollary that the size of any connected component of the Markoff mod graph is divisible by . In combination with the work of Bourgain, Gamburd, and Sarnak, Chen's result proves a conjecture of Baragar for all but finitely many primes: the Markoff mod graph is connected. In this note, we provide an alternative proof for the Markoff corollary of Chen's theorem.

Paper Structure

This paper contains 2 sections, 2 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:chen']}

Key Result

Theorem 1.2

Fix $\varepsilon>0$. For sufficiently large $p$, there is a connected component $\mathcal{C}_p$ of $\mathcal{G}_p$ for which $|\mathcal{G}_p \backslash \mathcal{C}_p| < p^\varepsilon$. Furthermore, the set of primes for which $\mathcal{G}_p$ is connected has natural density 1.

Theorems & Definitions (4)

  • Conjecture 1.1: Baragar baragar
  • Theorem 1.2: Bourgain--Gamburd--Sarnak BGSbourgain
  • Theorem 1.3: Special case/corollary of Chen chen
  • proof