Extremal Kleinian groups
William Abikoff, William James Harvey · Proceedings of the American Mathematical Society · 2011
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.