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research article
Experiments with the Markoff Surface
We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and Baragar on strong approximation for the Markoff surface modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets . In the degenerate case of the Cayley cubic, we give a complete description of the orbits.
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Type
research article
Web of Science ID
WOS:000506504200001
Authors
Publication date
2022
Publisher
Published in
Volume
31
Issue
3
Start page
814
End page
829
Peer reviewed
REVIEWED
EPFL units
Available on Infoscience
March 3, 2020