Lifting Möbius groups.
If the monodromy representation of a VHS over a hyperbolic curve stabilizes a rank two subspace, there is a single non-negative Lyapunov exponent associated with it. We derive an explicit formula using only the representation in the case when the monodromy is discrete.
This is a small survey paper about connections between the arithmetic and geometric properties in the case of arithmetic Fuchsian groups.