A new proof of Chen’s theorem for Markoff graphs

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2025年6月20日

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Daniel E. Martin

摘要

In 2021, Chen proved that the size of any connected component of the Markoff mod $p$ $p$ graph is divisible by $p$ $p$ . 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 $p$ $p$ 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.