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A Note on Complex-Hyperbolic Kleinian Groups

A Note on Complex-Hyperbolic Kleinian Groups Let Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varGamma $$\end{document} be a discrete group of isometries acting on the complex hyperbolic n-space HCn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n_\mathbb {C}$$\end{document}. In this note, we prove that if Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varGamma $$\end{document} is convex-cocompact, torsion-free, and the critical exponent δ(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta (\varGamma )$$\end{document} is strictly lesser than 2, then the complex manifold HCn/Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n_\mathbb {C}/\varGamma $$\end{document} is Stein. We also discuss several related conjectures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Arnold Mathematical Journal Springer Journals

A Note on Complex-Hyperbolic Kleinian Groups

Arnold Mathematical Journal , Volume 6 (3-4) – Jun 8, 2020

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Publisher
Springer Journals
Copyright
Copyright © Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2020
ISSN
2199-6792
eISSN
2199-6806
DOI
10.1007/s40598-020-00143-x
Publisher site
See Article on Publisher Site

Abstract

Let Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varGamma $$\end{document} be a discrete group of isometries acting on the complex hyperbolic n-space HCn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n_\mathbb {C}$$\end{document}. In this note, we prove that if Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varGamma $$\end{document} is convex-cocompact, torsion-free, and the critical exponent δ(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta (\varGamma )$$\end{document} is strictly lesser than 2, then the complex manifold HCn/Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {H}^n_\mathbb {C}/\varGamma $$\end{document} is Stein. We also discuss several related conjectures.

Journal

Arnold Mathematical JournalSpringer Journals

Published: Jun 8, 2020

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