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Extremal Kleinian groups


Authors: William Abikoff and William J. Harvey
Journal: Proc. Amer. Math. Soc. 140 (2012), 267-278
MSC (2010): Primary 30F40; Secondary 20H15
DOI: https://doi.org/10.1090/S0002-9939-2011-10923-7
Published electronically: May 26, 2011
MathSciNet review: 2833539
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Abstract: In 1967, Lipman Bers proved his area inequalities for Kleinian groups and gave examples to show that they are sharp; a group for which equality holds is termed extremal. Maskit's work on function groups published during the next decade contained implicitly a characterization of all extremal groups for the second inequality.

Here we determine the class of extremal groups for the first area inequality: these maximal area groups are all torsion-free Schottky or almost Schottky groups. For completeness, we also show that any extremal group for the second area inequality is either quasi-Fuchsian or a regular b-group.


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Additional Information

William Abikoff
Affiliation: Department of Mathematics, University of Connecticut at Storrs, Storrs, Connecticut 06269-3009
Email: abikoff@math.uconn.edu

William J. Harvey
Affiliation: Department of Mathematics, King’s College, Strand, London, WC2R-2LS England
Email: Bill.Harvey@kcl.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2011-10923-7
Received by editor(s): May 14, 2010
Received by editor(s) in revised form: November 13, 2010
Published electronically: May 26, 2011
Communicated by: Michael Wolf
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.