Distribution of holonomy about closed geodesics in a product of hyperbolic planes (Q2851608)

In the spirit of the analogy between prime numbers and primitive closed geodesics, \textit{W. Parry} and \textit{M. Pollicott} [Ergodic Theory Dyn. Syst. 6, 133--148 (1986; Zbl 0626.58006)] showed that on a manifold, for which the geodesic flow is topologically mixing, the holonomy classes become equidistributed in \(\mathrm{SO}(d-1)\) with respect to the Haar measure as the lengths of the geodesics tend to infinity. In particular, this is true for manifolds with constant negative curvature, that is, the rank one locally symmetric spaces \(\Gamma \backslash \mathrm{SO}_0(d, 1)/\mathrm{SO}(d)\).NEWLINENEWLINEThis paper is devoted to a similar question for the higher rank group \(G=\mathrm{PSL}(2,\mathbb R)^{n+1}\). That is, the author studies the equidistribution of holonomy classes of closed geodesics on the manifold \(\Gamma \backslash G/K=\Gamma \backslash \mathcal H^{(n)}\), where \(\mathcal H^{(n)}=\mathbb H_0\times \mathbb H_1\times\dots\times\mathbb H_n\) is a product of \(n+1\) hyperbolic planes, \(K=\mathrm{PSO}(2)^{n+1}\) and \(\Gamma\subset \mathrm{PSL}(2,\mathbb R)^{n+1}\) is an irreducible cocompact lattice.