Kesten–McKay law for the Markoff surface mod p

Abstract

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.Résumé. -Pour chaque nombre premier p, on décrit les valeurs propres d'un graphe 3-régulier ayant environ p 2 sommets construit à partir de la surface de Markoff. On montre qu'elles suivent approximativement la loi de Kesten-McKay, qui décrit également les valeurs propres d'un graphe aléatoire régulier. On utilise la méthode des moments et l'action de GL 2 (Z) sur la surface de Markoff.

Non-planarity of Markoff graphs $\bmod~p$

“…For k ¤ 0 or 4, the number of fixed points is dictated by a curve (2.6) of genus 1, which in our case degenerates to a conic. Theorem 2.4 (de Courcy-Ireland and Magee [13]). There is an absolute constant C > 0 such that any reduced word of length L in m 1 , m 2 , m 3 has at most C L p fixed points.…”

Polynomial Equations in Subgroups and Applications

“…Bourgain, Gamburd and Sarnak [2,3] have obtained several major results towards Conjecture 1.1, see also [4,6,7,10]. For example, by [2, Theorem 1] we have (1.3) # (M p \ C p ) = p o (1) , as p → ∞, and also by [2,Theorem 2] we know that Conjecture 1.1 holds for all but maybe at most X o (1) primes p ≤ X as X → ∞.…”

Non‐planarity of SL(2,Z)$\operatorname{SL}(2,\mathbb {Z})$‐orbits of origamis in H(2)$\mathcal {H}(2)$

“…Indeed, it follows from the separator theorem for planar graphs of Lipton-Tarjan [8] that planar graphs cannot form an expander family. We direct the reader to the exposition of this fact in de Courcy-Ireland's recent work [2,Section 12] demonstrating that, for certain primes 𝑝, Markoff graphs modulo 𝑝 -another conjectured expander family -are non-planar.…”