Poincar� series and holomorphic averaging

Amenability, Poincar E Series and Quasiconformal Maps

Any covering Y ! X of a hyperbolic Riemann surface X of-nite area determines an inclusion of Teichm uller spaces Teich(X) ,! Teich(Y). We show this map is an isometry for the Teichm uller metric if the covering is amenable, and contracting otherwise. In particular , we establish jjjj < 1 for classical Poincar e series (Kra's `Theta conjecture'). The appendix develops the theory of geometric lim...

Poincar E Series and Comparison Theorems for Variable Negative Curvature

Poincaré series have always played a central rôle in the theory of automorphic functions and harmonic analysis on manifolds of constant negative curvature. The trace formulae approach was introduced by Selberg [20] in 1956 and has become the key to their analysis. However, with the advent of a more geometrical viewpoint (through the work of Patterson [15, 16], Sullivan [21, 22] and others) it b...

Universality of Holomorphic Discrete Series

The goal here is to recover an apocryphal result on the structure of holomorphic discrete series representations of symplectic groups Spn(R) and unitary groups U(p, q) for sufficiently high highest weight of the lowest K-type. The same sort of argument applies to other groups of hermitian type, for example the classical groups O(n, 2) and O∗(2n). For Sp(n), the maximal compact is isomorphic to ...

Legendrian distributions with applications to relative Poincar series

A Toric Ring with Irrational Poincar E-betti Series

{ We show that there exists a toric curve in P 8 , whose homogeneous coordinate ring has a presentation with 12 quadratic relations and whose Poincar e-Betti series is irrational. The example was found by a computer search, aiming at a homological classiication of those toric curves that have a quadratic presentation in P n?1 for n 9. Some other consequences of this search are also presented. U...