VOLUME GROWTH AND CLOSED GEODESICS ON RIEMANNIAN MANIFOLDS OF HYPERBOLIC TYPE

We study the volume growth function of geodesic spheres in the universal Riemannian covering of a compact manifold of hyperbolic type. Furthermore, we investigate the growth rate of closed geodesics in compact manifolds of hyperbolic type

Fiberwise volume growth via Lagrangian intersections

We consider Hamiltonian diffeomorphisms $\phi$ of the unit cotangent bundle over a closed Riemannian manifold $(M,g)$ which extend to Hamiltonian diffeomorphisms of $T^*M$ equal to the time-1-map of the geodesic flow for $|p| \ge 1$. For such diffeomorphisms we establish uniform lower bounds for the fiberwise volume growth of $\phi$ which were previously known for geodesic flows and which depend only on $(M,g)$ or on the homotopy type of $M$. More precisely, we show that for each $q \in M$ the volume growth of the unit ball in $T_q^*M$ under the iterates of $\phi$ is at least linear if $M$ is rationally elliptic, is exponential if $M$ is rationally hyperbolic, and is bounded from below by the growth of the fundamental group of $M$. In the case that all geodesics of $g$ are closed, we conclude that the slow volume growth of every symplectomorphism in the symplectic isotopy class of the Dehn--Seidel twist is at least 1, completing the main result of \cite{FS:GAFA}. The proofs use the Lagrangian Floer homology of $T^*M$ and the Abbondandolo--Schwarz isomorphism from this homology to the homology of the based loop space of $M$.Comment: 19 pages, latex2

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

Arithmetic lattices and weak spectral geometry

This note is an expansion of three lectures given at the workshop "Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces" held at Kyoto University in December of 2006 and will appear in the proceedings for this workshop.Comment: To appear in workshop proceedings for "Topology, Complex Analysis and Arithmetic of Hyperbolic Spaces". Comments welcom