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Abstract
We will examine the Poincare Theta series and its behavior under geometric convergence of Kleinian groups. We first define automorphic forms and Poincare Theta series for Kleinian groups acting on the unit ball. As a result, we construct nonconstant cross sections of vector bundles over hyperbolic 3-orbifolds. Then we prove that when a sequence of Kleinian groups acting on $\Delta\sp3$ converges geometrically, the corresponding Poincare Theta series converges locally and uniformly in $\Delta\sp3$. We also prove a similar theorem for Kleinian groups acting on C. Applying these theorems, we show that the genus and hyperbolic area of the quotient Riemann surfaces induced by the geometric limit group cannot be increased by passage to a geometric limit. We explicitly construct a sequence of strongly convergent Kleinian groups and calculate their Poincare Theta series which turns out to be related to the Riemann $\zeta$-function.
