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Lyapunov spectrum of ball quotients with applications to commensurability questions. (English) Zbl 1334.22010

Let \(\mathcal B^n\) be the complex ball \( \{z = (z_1, \dots , z_n) : \Sigma |z_i|^2 < 1 \}\) and \(B\) be a ball quotient, i.e. the quotient of \(\mathcal B^n\) by some lattice in the group \(\mathrm{PU}(1, n)\). The problem of commensurability of such lattices is investigated. Some special technique (Lyapunov exponents and Lyapunov spectrum) is developed and used for finding invariants of commensurability for such lattices and for the corresponding balls \(B\).
The Lyapunov exponents measure the logarithmic growth rate of the cohomology classes under parallel transport along the geodesic flow of the ball quotient. The Lyapunov spectrum is one of the characteristics of Lyapuniov exponents used in the theory of dynamic systems. In this paper, the Lyapunov spectrum of ball quotients arising from cyclic coverings is investigated.
As a corollary, a classification of the commensurability classes of all non-arithmetic ball quotients (for \(n=2\)) known to authors is obtained.

MSC:

22E40 Discrete subgroups of Lie groups
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