Residual Transitivity implies Minimality for Markoff Surfaces over $p$-adic Integers, by Means of $p$-adic Flows

TL;DR

This paper offers a new proof that modulo p transitivity by automorphisms ensures minimality of p-adic points on Markoff surfaces, extending previous results to additional parameter cases using p-adic flow techniques.

Contribution

Provides an alternative proof linking transitivity and minimality on Markoff surfaces and generalizes to more parameters using p-adic analytic flows.

Findings

Modulo p transitivity implies minimality of p-adic points.

New proof technique using p-adic flows.

Extension to parameters D congruent to 0 mod p^2 or with (D-4) quadratic residue.

Abstract

Let $X_D^\ast$ be the non-singuar locus of the Markoff surface $X_D\colon x^2+y^2+z^2=xyz+D$ and consider the set of its $p$-adic integer points $X_D^\ast(\mathbb{Z}_p)$. It is known to Bourgain, Gamburd, and Sarnak that the modulo $p$ transitivity by algebraic automorphisms of $X_0^\ast$ implies minimality of $X_0^\ast(\mathbb{Z}_p)$ by algebraic automorphisms. In this paper, we provide an alternative proof of this fact, by some techniques to study $p$-adic analytic flows. This establish a slight generalization to those parameters $D$ congruent to $0$ modulo $p^2$ or $(D-4)$ being a nonzero quadratic residue.