A new proof of Chen’s theorem for Markoff graphs

Abstract In 2021, Chen proved that the size of any connected component of the Markoff mod $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> graph is divisible by $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> . In combination with the work of Bourgain, Gamburd, and Sarnak, Chen’s result resolves a conjecture of Baragar for all but finitely many primes: the Markoff mod $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> graph is connected. In particular, strong approximation for Markoff triples holds for all but finitely many primes. We provide an alternative proof of Chen’s theorem.