Lyapunov spectrum of ball quotients with applications to commensurability questions

Abstract

Abstract. We determine the Lyapunov spectrum of ball quotients arising from cyclic coverings. The computations are performed by rewriting the sum of Lyapunov exponents as ratios of intersection numbers and by the analysis of the period map near boundary divisors.As a corollary, we complete the classification of commensurability classes of all presently known non-arithmetic ball quotients.

New non-arithmetic complex hyperbolic lattices

“…The table for Mostow and Deligne-Mostow groups show that there are at most 13 commensurability classes of Deligne-Mostow lattices in PU(2, 1). As mentioned above, the results in [6], [16] and [19] imply that there are in fact precisely 9 commensurability classes there. 7.2.1.…”

Parabolic Degrees and Lyapunov Exponents for Hypergeometric Local Systems

“…Soon this link was observed in other settings. In [KM16] it was used as a new invariant to classify hyperbolic structures and distinguish Deligne-Mostow's non-arithmetic lattices in SL 2 (C). In [Fil14] a similar formula was observed for higher weight variation of Hodge structures.…”

Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular Embeddings

“…This has already been remarked by McMullen [21,Section 10], and can in fact be generalized to ball quotients [17,Theorem 5.4]. We recall the arguments for the convenience of the reader.…”

“…In this situation, the period maps are Schwarz triangle maps, the monodromy is a possibly indiscrete triangle group, and the Lyapunov exponents are quotients of areas of hyperbolic triangles. Other examples, where individual Lyapunov exponents have been obtained by computing the degrees of line bundles, are the Veech-Ward-Bouw-Möller-Teichmüller curves [3], [29], cyclic covers of P 1 [9] and more generally Deligne-Mostow ball quotients [17]. Modular embeddings of H into a product H k have been studied e. g. in [5] for the action of a Schwarz triangle group on the left and the direct product of its Galois conjugates on the right (where k is the degree of the trace field over Q) or for non-arithmetic Teichmüller curves in [22], [21] for the action of the Veech group and its Galois conjugates.…”

“…One can even replace the Teichmüller flow by the geodesic flow on an arbitrary hyperbolic curve H /Γ (or more generally a ball quotient, see [17]) and an analogous formula will hold for the dynamical cocycle coming from the monodromy action of the fundamental group Γ on the cohomology of a family of curves φ : X → H /Γ (or more generally on a variation of Hodge structures (VHS) of weight one).…”