Primitive geodesic lengths and (almost) arithmetic progressions

In this article, we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every non-compact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma

Effects of Disks on Gravitational Lensing by Spiral Galaxies

Gravitational lensing of a quasar by a spiral galaxy should often be accompanied by damped Lyman-alpha absorption and dust extinction due to the intervening gaseous disk. In nearly edge-on configurations, the surface mass density of the gas and stars in the disk could by itself split the quasar image and contribute significantly to the overall lensing cross section. We calculate the lensing probability of a disk+halo mass model for spiral galaxies, including cosmic evolution of the lens parameters. A considerable fraction of the lens systems contains two images with sub-arcsecond separation, straddling a nearly edge-on disk. Because of that, extinction by dust together with observational selection effects (involving a minimum separation and a maximum flux ratio for the lensed images), suppress the detection efficiency of spiral lenses in optical wavebands by at least an order of magnitude. The missing lenses could be recovered in radio surveys. In modifying the statistics of damped Lyman-alpha absorbers, the effect of extinction dominates over the magnification bias due to lensing.Comment: 19 pages, 12 figures; submitted to Ap

On the monodromy of the Hitchin connection

For any genus g > 1 we give an example of a family of smooth complex projective curves of genus g such that the image of the monodromy representation of the Hitchin connection on the sheaf of generalized SL(2)-theta functions of level l different from 1,2,4 and 8 contains an element of infinite order.Comment: 20 pages, revised versio