When do quotients of bounded domains contain closed Riemann surfaces?

Question

Let $D$ be a bounded domain in $\mathbb{C}^n$, and let $\gamma$ be a biholomorphism of $D$ such that for all $\epsilon>0$ there is a point $z\in D$ such that the Kobayashi distance from $z$ to $\gamma(z)$ is at most $\epsilon$ (and $\gamma$ is not the identity).

Is it ever possible to have a non-constant holomorphic map from a closed Riemann surface to $D/\langle \gamma \rangle$?

For example, if $D$ is the unit disc in $\mathbb{C}$, then $D/\langle \gamma \rangle$ is biholomorphic to the punctured disc, so any holomorphic map from a closed Riemann surface to $D/\langle \gamma \rangle$ must be constant.

Answer 1

Proposition. If $D$ is a bounded connected domain in ${\mathbb C}^n$ and $\Gamma$ is a nilpotent group of biholomorphic transformations of $D$ acting freely and properly discontinuously on $D$, then every map from a compact Riemann surface to $D/\Gamma$ is constant.

Proof. I will use the following theorem, which is a very special case of a theorem due to Lyons and Sullivan, Theorem 1 in T. Lyons and D. Sullivan, Function theory, random paths, and covering spaces, J. Diff. Geom., 19 (1984), 299-323:

Suppose that $S$ is a compact connected Riemann surface, $R\to S$ is a regular cover with nilpotent deck-transformation group. Then every bounded holomorphic function on $R$ is constant. [Lyons and Sullivan proved this for harmonic functions on Riemannian manifolds; I am pretty sure this theorem was known for Riemann surfaces much earlier, I just do not know a better reference.]

Now, in our situation, suppose that $S$ is a compact connected Riemann surface, $f: S\to M=D/\Gamma$ is a holomorphic function. Then $f$ lifts to a holomorphic function $F: R\to D$, where $p: R\to S$ is the regular cover associated with the natural homomorphism $\pi_1(S)\to \pi_1(M)\to \Gamma$. The deck-transformation group of $p$ is the image of the homomorphism $\pi_1(S)\to \Gamma$ and, hence, nilpotent. By the theorem of Lyons and Sullivan above, since $D$ is a bounded domain, $F$ has to be a constant function, hence, $f$ is constant as well. qed.

Incidentally, one can also ask for conditions under which quotients $D/\Gamma$ are Stein, where $\Gamma$ is infinite cyclic. (This is not always the case.) The central conjecture here is the one due to Alan Huckleberry that if $D$ is a bounded contractible domain of holomorphy and $\Gamma$ is a subgroup of 1-parameter group of biholomorphic automorphisms of $D$, then $D/\Gamma$ is Stein. You can find partial results and many references in the habilitation thesis of Christian Miebach:

Discrete quotients of Stein manifolds, the geometry of holomorphic Lie group actions, and domains in complex homogeneous spaces.