On discrete subgroups of the complex unit ball

Abstract

In this paper, we study conditions for a discrete subgroup of the automorphism group of the $n$-dimensional complex unit ball to be of convergence type or second kind, connecting these classifications to the existence of Green's functions and subharmonic or harmonic functions on its quotient space. Furthermore, we extend the definitions of convergence and divergence types to bounded symmetric domains, introducing a Poincaré series and providing a new criterion for discrete subgroups acting on these domains.