Lyapunov exponents of non-arithmetic complex hyperbolic lattices

To a flat vector bundle over a Riemannian manifold, one can associate its Lyapunov exponents, the logarithmic growth rates of sections when parallel transported along the geodesic flow. Flat bundles occurring in nature are the relative cohomology bundles associated to families of curves, or more generally Kaehler manifolds. In the case of a family of curves over a hyperbolic curve, there is a beautiful formula, first discovered by Kontsevich, that relates the sum of Lyapunov exponents to the degrees of certain line bundles. I will discuss a variant of this formula, where the base is a ball quotient, the orbit space of a lattice acting on complex hyperbolic n-space. The most prominent examples of non-arithmetic complex hyperbolic lattices were found by Picard, Terada, Deligne, Mostow and Thurston. Their ball quotients parametrize cyclic coverings of the line and thus come naturally with a flat vector bundle, whose Lyapunov exponents we can compute. Moreover, the Lyapunov exponents can be used to distinguish commensurability classes in this case. Joint work with M. Möller.

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