Super Strong Approximation for Markoff modulo-p graphs

For each prime $p$, let $\mathcal{G}_p$ be the corresponding Markoff modulo-$p$ graph. Super Strong Approximation. The collection of Markoff modulo-$p$ graphs $(\mathcal{G}_p)_p$ forms an expander family.

This is the spectral-expansion formulation of super-strong approximation for the Markoff graphs and is used in the paper to obtain uniform bounds for lifts along shortest paths. It is explicitly described as expected, with the source pointing to experimental evidence; the conjecture remains open.