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Distribution of holonomy about closed geodesics in a product of hyperbolic planes
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 6, December 2012
- pp. 1613-1653
- 10.1353/ajm.2012.0042
- Article
- Additional Information
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.