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The Patterson–Sullivan measure and proper conjugation for Kleinian groups of divergence type

Published online by Cambridge University Press:  01 April 2009

KATSUHIKO MATSUZAKI
Affiliation:
Department of Mathematics, Okayama University, Okayama 700-8530, Japan (email: matsuzak@math.okayama-u.ac.jp)
YASUHIRO YABUKI
Affiliation:
Mathematical Institute, Tohoku University, Aoba-ku, Sendai 980-8578, Japan (email: sa3m30@math.tohoku.ac.jp)

Abstract

A Kleinian group (a discrete subgroup of conformal automorphisms of the unit ball) G is said to have proper conjugation if it contains the conjugate αGα−1 by some conformal automorphism α as a proper subgroup in it. We show that a Kleinian group of divergence type cannot have proper conjugation. Uniqueness of the Patterson–Sullivan measure for such a Kleinian group is crucial to our proof.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2008

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