The cycle structure of a Markoff automorphism over finite fields

Abstract

We begin an investigation of the action of pseudo-Anosov elements of Out(F2) on the Markoff-type varietiesXκ : x 2 + y 2 + z 2 = xyz + 2 + κ over finite fields Fp with p prime. We first make a precise conjecture about the permutation group generated by Out(F2) on X−2(Fp) that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting 'generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov g ∈ Out(F2), there is always an orbit of g of length ≥ C log p + O(1) on Xκ(Fp) where C > 0 is given in terms of the eigenvalues of g viewed as an element of GL2(Z). This improves on a result of Silverman from [24] that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic (p → ∞) behavior of the longest orbit of a fixed pseudo-Anosov g acting on X−2(Fp) is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak [21]. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture in Conjecture 1.10.

Orbits on K3 Surfaces of Markoff Type

“…We note that an orbit containing points of this form does not fit into the family described in Remark 10.8, but this does not preclude it coming from some other characteristic 0 orbit, so we continue analyzing the present example. In particular, we see that W 11 (F 53 ) contains the points (38, −38, 1) (15,38,12) (15,11,12) 11,12). This suggests that we should take a point (α, −α, 1) ∈ W k satisfying…”

“…• Given a pseudo-Anosov element g ∈ Out(F 2 ), g induces a permutation g p on M 1,k (F p ) for each prime p. Cerbu-Gunther-Magee-Peilen [12] prove that asymptotically, the action of g p on M 1,k (F p ) has an orbit of size at least log(p) log |λ| + O g (1), where λ is the eigenvalue of largest modulus of g when viewed as an element of GL 2 (Z).…”

Kesten–McKay law for the Markoff surface mod p

“…In this case, we use the following bound of Cerbu-Gunther-Magee-Peilen ([CGMP20, Lemma 3.9]). We refer to [CGMP20] for the proof. The assumption that all the entries have absolute value at least 2 makes it possible to implement a rigorous version of the heuristic in Section 3.…”

Experiments with the Markoff Surface

“…There is one k for each p that generates a Markoff graph with an especially large number of components. In Table 4.1, these pairs (p, k) are (5, 1), (7,2), (11,9), (13,12) . .…”

“…They prove this for p ≡ 3 mod 4 as well, but this requires an additional hypothesis about p. Carmon shows in the appendix to [19] that this assumption holds except for another sparse sequence of primes. It had been conjectured around the same time by Cerbu-Gunther-Magee-Peilen that the group is alternating for p ≡ 3 mod 16 and symmetric otherwise [7]. This phenomenon of being fully transitive is a kindred spirit to expansion: Not only is the graph/action connected/transitive, but it is very robustly connected so that there are many ways to go from one point to any other.…”