A new proof of Chen's theorem for Markoff graphs

TL;DR

This paper offers an alternative proof of Chen's theorem, demonstrating that the Markoff mod p graph is connected for all but finitely many primes, building on prior work by Chen and others.

Contribution

It provides a new proof of Chen's theorem regarding the connectivity of Markoff graphs, simplifying and extending previous results.

Findings

Markoff mod p graph is connected for all but finitely many primes

Chen's theorem confirms the divisibility of component sizes by p

Alternative proof enhances understanding of Markoff graph properties

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.