Non-planarity of Markoff graphs mod p
arXiv:2105.12411
Abstract
We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to the Markoff equation modulo prime numbers greater than 7. The proof uses Euler characteristic and an enumeration of the short cycles in these graphs. Non-planarity for large primes would follow assuming a spectral gap, which was the original motivation. For primes congruent to 1 modulo 4, or congruent to 1, 2, or 4 modulo 7, explicit constructions give an alternate proof of non-planarity.
v4: 32 pages, 10 figures; includes proof of the conjecture from v2; accepted for publication in Commentarii Mathematici Helvetici