Distribution of holonomy about closed geodesics in a product of hyperbolic planes

Details Distribution of holonomy about closed geodesics in a product of hyperbolic planes Distribution of holonomy about closed geodesics in a product of hyperbolic planes

2009-11-02

arxiv.org/abs/0911.0329v2

Mathematics

Number Theory

42 pages. Revised title. Theorem 3 is strengthened. To appear in Amer. J. Math

Scientific paper

Let $\calM=\Gamma\bs \calH^{(n)}$, where $\calH^{(n)}$ is a product of $n+1$ hyperbolic planes and $\Gamma\subset\PSL(2,\bbR)^{n+1}$ is an irreducible cocompact lattice. We consider closed geodesics on $\calM$ that propagate locally only in one factor. We show that, as the length tends to infinity, the holonomy rotations attached to these geodesics become equidistributed in $\PSO(2)^n$ with respect to a certain measure. For the special case of lattices derived from quaternion algebras, we can give another interpretation of the holonomy angles under which this measure arises naturally.

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