A new proof of Chen's theorem for Markoff graphs

Daniel E. Martin

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arXiv · 2502.15960

A new proof of Chen's theorem for Markoff graphs

Daniel E. Martin

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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.

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Daniel E. Martin. 2025-02-21. A new proof of Chen's theorem for Markoff graphs. https://arxiv.org/abs/2502.15960

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