Mixing of frame flow for rank one locally symmetric spaces and measure classification

Curvature Characterization and Classification of Rank-one Symmetric Spaces

Symbolic Dynamics for the Geodesic Flow on Locally Symmetric Orbifolds of Rank One

We present a strategy for a geometric construction of cross sections for the geodesic flow on locally symmetric orbifolds of rank one. We work it out in detail for Γ\H, where H is the upper half plane and Γ = PΓ0(p), p prime. Its associated discrete dynamical system naturally induces a symbolic dynamics on R. The transfer operator produced from this symbolic dynamics has a particularly simple s...

Cohomogeneity One Actions on Noncompact Symmetric Spaces of Rank One

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces CHn, n ≥ 3. For the quaternionic hyperbolic spaces HHn, n ≥ 3, we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classificat...

On the number of ends of rank one locally symmetric spaces

Let Y be a noncompact rank one locally symmetric space of finite volume. Then Y has a finite number e(Y ) > 0 of topological ends. In this paper, we show that for any n ∈ N, the Y with e(Y ) ≤ n that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant cn such that n-cusped arithmetic orbifolds do not exist in dimension greater than cn. We make thi...

L-Spectral theory of locally symmetric spaces with Q-rank one

We study the L-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one.