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

TL;DR

This paper studies the distribution of holonomy rotations along certain closed geodesics in a product of hyperbolic planes, showing they become equidistributed in a specific measure as geodesic length increases.

Contribution

It proves the equidistribution of holonomy rotations for geodesics in a product of hyperbolic planes, including a special case for quaternion algebra lattices.

Findings

Holonomy rotations become equidistributed as geodesic length increases.

The distribution is described by a specific measure on ^n.

Special interpretation for quaternion algebra-derived lattices.

Abstract

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.