Diophantine geometry of character varieties

We consider moduli spaces of special linear rank two local systems on a punctured surface with prescribed monodromy traces around the punctures. We discuss a proof that every such moduli space is log Calabi-Yau, and relate the property to a symmetry of generating series for counts of disjoint simple curves on the surface. Time permitting, we discuss new Diophantine results on these moduli spaces suggested by the global geometry, generalizing work of Markoff.