Stable reduction and large Nielsen orbits on generalized Markoff surfaces

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = τ+2$. We show that outside of the reductions of a certain exceptional finite set of $\overline{\mathbb{Q}}$-points foun