Extremal Kleinian groups

Abikoff, William; Harvey, William

doi:10.1090/S0002-9939-2011-10923-7

Extremal Kleinian groups
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by William Abikoff and William J. Harvey PDF

Proc. Amer. Math. Soc. 140 (2012), 267-278 Request permission

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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