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

Let $\text{calM}=\mathrm{\Gamma }\text{bs}{\text{calH}}^{(n)}$, where ${\text{calH}}^{(n)}$ is a product of $n+1$ hyperbolic planes and $\mathrm{\Gamma }\subset \text{PSL}(2,\text{bbR}{)}^{n+1}$ is an irreducible cocompact lattice. We consider closed geodesics on $\text{calM}$ 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 $\text{PSO}(2{)}^n$ 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.