Ergodic Theory and Dynamical Systems

Research Article

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

KATSUHIKO MATSUZAKIa1 and YASUHIRO YABUKIa2

a1 Department of Mathematics, Okayama University, Okayama 700-8530, Japan (email: matsuzak@math.okayama-u.ac.jp)

a2 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.

(Received October 09 2007)

(Revised February 26 2008)