Kesten–McKay law for the Markoff surface mod p
Annales Henri Lebesgue, Volume 4 (2021), pp. 227-250.


Keywords Markoff surface, Kesten–McKay law, cubic surfaces, graphs and groups


For each prime p, we study the eigenvalues of a 3-regular graph on roughly p 2 vertices constructed from the Markoff surface. We show they asymptotically follow the Kesten–McKay law, which also describes the eigenvalues of a random regular graph. The proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.


[Aig13] Aigner, Martin Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings, Springer, 2013 | Zbl

[Bar91] Baragar, Arthur The Markoff equation and equations of Hurwitz, Ph. D. Thesis, Brown University, USA (1991) | MR

[BGS16] Bourgain, Jean; Gamburd, Alexander; Sarnak, Peter Markoff Surfaces and Strong Approximation: 1 (2016) (

[Car57] Carlitz, Leonard The number of points on certain cubic surfaces over a finite field, Boll. Unione Mat. Ital., Volume 12 (1957), pp. 19-21 | MR | Zbl

[CGMP20] Cerbu, Alois; Gunther, Elijah; Magee, Michael; Peilen, Luke The cycle structure of a Markoff automorphism over finite fields, J. Number Theory, Volume 211 (2020), pp. 1-27 | DOI | MR | Zbl

[CL09] Cantat, Serge; Loray, Frank Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation., Ann. Inst. Fourier, Volume 59 (2009) no. 7, pp. 2927-2978 | DOI | Numdam | Zbl

[ET48a] Erdős, Pál; Turán, Pál On a problem in the theory of uniform distribution. I, Proc. Akad. Wet. Amsterdam, Volume 51 (1948), pp. 1146-1154 | MR | Zbl

[ET48b] Erdős, Pál; Turán, Pál On a problem in the theory of uniform distribution. II, Proc. Akad. Wet. Amsterdam, Volume 51 (1948), pp. 1262-1269 | MR | Zbl

[FK65] Fricke, Robert; Klein, Felix Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bände 3, 4, Johnson Reprint Corp.; Teubner, 1965

[Fri96] Fricke, Robert Über die Theorie der automorphen Modulgrupper, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 1896 (1896), pp. 91-101

[GJS99] Gamburd, Alexander; Jakobson, Dmitry; Sarnak, Peter Spectra of elements in the group ring of SU(2), J. Eur. Math. Soc., Volume 1 (1999) no. 1, pp. 51-85 | DOI | MR | Zbl

[Kes59] Kesten, Harry Symmetric random walks on groups, Trans. Am. Math. Soc., Volume 92 (1959), pp. 336-354 | DOI | MR | Zbl

[KMSV20] Konyagin, Sergeĭ V.; Makarychev, Sergey V.; Shparlinski, Igor E.; Vyugin, Ilya V. On the Structure of Graphs of Markoff Triples, Q. J. Math., Volume 71 (2020) no. 2, pp. 637-648 | DOI | MR | Zbl

[Mar80] Markoff, Andreĭ Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 17 (1880) no. 3, pp. 379-399 | DOI | Zbl

[McK81] McKay, Bredan D. The expected eigenvalue distribution of a large regular graph, Linear Algebra Appl., Volume 40 (1981), pp. 203-216 | DOI | MR

[MKS04] Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Dover Publications, 2004 | Zbl

[Mon94] Montgomery, Hugh L. Ten lectures on the interface between analytic number theory and harmonic analysis, Regional Conference Series in Mathematics, 84, American Mathematical Society, 1994 | MR | Zbl

[MP18] Meiri, Chen; Puder, Doron and The Markoff Group of Transformations in Prime and Composite Moduli, Duke Math. J., Volume 167 (2018) no. 14, pp. 2679-2720 | DOI | MR | Zbl

[Nie17] Nielsen, Jakob Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann., Volume 78 (1917), pp. 385-397 | DOI | MR | Zbl

[Sel91] Selberg, Atle Collected Papers. Vol. II, Springer, 1991 (Lectures on sieves, p. 65–247) | Zbl

[Vaa85] Vaaler, Jeffrey D. Some extremal functions in Fourier analysis, Bull. Am. Math. Soc., Volume 12 (1985) no. 2, pp. 183-216 | DOI | MR | Zbl

[ÈH74] Èl’-Huti, M. H. Cubic surfaces of Markov type, Math. USSR, Sb., Volume 22 (1974) no. 3, pp. 333-348 (translated by R. Lenet) | DOI | MR | Zbl