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

Abstract

Let ${\mathcal M}=\Gamma\backslash {\mathcal H}^{(n)}$, where ${\mathcal H}^{(n)}$ is a product of $n+1$ hyperbolic planes and $\Gamma\subset{\rm PSL}(2, {\Bbb R})^{n+1}$ is an irreducible cocompact lattice. We consider closed geodesics on ${\mathcal M}$ 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 ${\rm PSO}(2)^n$ 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.