Properly discontinuous actions on bounded domains

TL;DR

This paper establishes conditions under which quotients of bounded domains with properly discontinuous free actions are Stein manifolds, with specific results for cyclic actions on the unit ball and bidisc.

Contribution

It provides new criteria using Poincaré series and limit sets to determine when such quotients are Stein, extending understanding of complex geometric group actions.

Findings

Quotients of cyclic free properly discontinuous actions on the unit ball are Stein.

Quotients of cyclic free properly discontinuous actions on the bidisc are Stein.

Sufficient conditions involve Poincaré series and orbit limit sets.

Abstract

We give sufficient conditions for the quotient of a free, properly discontinuous action on a bounded domain of holomorphy to be a Stein manifold in terms of Poincar\'e series or limit sets for orbits. An immediate consequence is that the quotient of any cyclic, free, properly discontinuous action on the unit ball or the bidisc is Stein.