Sur les quotients discrets de semi-groupes complexes

TL;DR

This paper investigates the conditions under which quotients of certain complex semi-groups associated with Hermitian symmetric spaces are Stein manifolds, providing new insights and counterexamples to previous conjectures.

Contribution

It establishes a sufficient condition for the quotient to be Stein and shows that in general, the quotient is not Stein, disproving a prior conjecture.

Findings

Provided a criterion for Stein quotients of semi-group actions.

Disproved the conjecture that all such quotients are Stein.

Showed that most quotients are not Stein, contrary to previous beliefs.

Abstract

Let $X=G/K$ be an irreducible Hermitian symmetric space of the non-compact type and let $S\in G^\mbb{C}$ be the associated compression semi-group. Let $\Gamma$ be a discrete subgroup of $G$. We give a sufficient condition for $\Gamma\backslash S$ to be a Stein manifold. Moreover, we show that in general $\Gamma\backslash S$ is not Stein, which disproves a conjecture by Achab, Betten and Kr\"otz.