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Holomorphic functions on geometrically finite quotients of the ball
[Fonctions holomorphes sur les quotients géométriquement finis de la boule]
William Sarem1
1Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France
Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 321-348

Let $\Gamma $ be a discrete and torsion-free subgroup of $\mathrm{PU}(n,1)$, the group of biholomorphisms of the unit ball in $\mathbb{C}^{n}$, denoted by $\mathbb{H}^{n}_{\mathbb{C}}$. We show that if $\Gamma $ is Abelian, then $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma $ is a Stein manifold. If the critical exponent $\delta (\Gamma )$ of $\Gamma $ is less than 2, a conjecture of Dey and Kapovich predicts that the quotient $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma $ is Stein. We confirm this conjecture in the case where $\Gamma $ is parabolic or geometrically finite. We also study the case of quotients with $\delta (\Gamma )=2$ that contain compact complex curves and confirm another conjecture of Dey and Kapovich. We finally show that $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma $ is Stein when $\Gamma $ is a parabolic or geometrically finite group preserving a totally real and totally geodesic submanifold of $\mathbb{H}^{n}_{\mathbb{C}}$, without any hypothesis on the critical exponent.

Soit $\Gamma $ un sous-groupe discret et sans torsion de $\mathrm{PU}(n,1)$, le groupe des biholomorphismes de la boule unité de $\mathbb{C}^{n}$, notée $\mathbb{H}^{n}_{\mathbb{C}}$. Nous montrons que si $\Gamma $ est abélien, alors $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma $ est une variété de Stein. Si l’exposant critique $\delta (\Gamma )$ de $\Gamma $ est inférieur à 2, une conjecture de Dey et Kapovich prédit que le quotient $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma $ est une variété de Stein. Nous confirmons cette conjecture dans le cas où $\Gamma $ est parabolique ou géométriquement fini. Nous étudions également le cas des quotients avec $\delta (\Gamma )=2$ qui contiennent des courbes complexes compactes, et confirmons une autre conjecture de Dey et Kapovich. Enfin, nous montrons que $\mathbb{H}^{n}_{\mathbb{C}}/\Gamma $ est de Stein lorsque $\Gamma $ est un groupe parabolique ou géométriquement fini préservant une sous-variété totalement réelle et totalement géodésique de $\mathbb{H}^{n}_{\mathbb{C}}$, sans aucune hypothèse sur l’exposant critique.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.328
Classification : 22E40, 32Q28
Keywords: Discrete subgroups, Stein manifolds, critical exponent, Patterson-Sullivan theory
Mots-clés : Sous-groupes discrets, variétés de Stein, exposant critique, théorie de Patterson-Sullivan

William Sarem  1

1 Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Holomorphic functions on geometrically~finite quotients of the ball},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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William Sarem. Holomorphic functions on geometrically finite quotients of the ball. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 321-348. doi: 10.5802/jep.328

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