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Discrete Groups on the Sphere at Infinity

Discrete Groups on the Sphere at Infinity PETER J. NICHOLLS We denote by G the group of Moebius transformations in R", n ^ 2, which preserve the unit ball B = {x : |x| < 1} and hence also the sphere S = {x : |x| = 1}. An element of G acts on S x S by the rule y{zy,z ) = (y{z),y{z)) 2 l 2 wher e y e G an d z,z,eS. l 2 In his recent important paper Sullivan [3] has explained the dynamical picture, in the context of Lebesgue measure, of the action of discrete subgroups of G both on S and on SxS. One could raise questions about the dynamical picture of a group action on S x S x.. . x S (m factors) for m > 2 and it is the purpose of this note to answer these questions. It seems to be accepted by many that such group actions will be uninteresting from an ergodic point of view—we will explain the precise extent to which this is true. We will begin by reviewing Sullivan's theorems and for this we need some preliminary definitions and notation. Suppose F is a discrete subgroup of G and we think of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Discrete Groups on the Sphere at Infinity

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/15.5.488
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Abstract

PETER J. NICHOLLS We denote by G the group of Moebius transformations in R", n ^ 2, which preserve the unit ball B = {x : |x| < 1} and hence also the sphere S = {x : |x| = 1}. An element of G acts on S x S by the rule y{zy,z ) = (y{z),y{z)) 2 l 2 wher e y e G an d z,z,eS. l 2 In his recent important paper Sullivan [3] has explained the dynamical picture, in the context of Lebesgue measure, of the action of discrete subgroups of G both on S and on SxS. One could raise questions about the dynamical picture of a group action on S x S x.. . x S (m factors) for m > 2 and it is the purpose of this note to answer these questions. It seems to be accepted by many that such group actions will be uninteresting from an ergodic point of view—we will explain the precise extent to which this is true. We will begin by reviewing Sullivan's theorems and for this we need some preliminary definitions and notation. Suppose F is a discrete subgroup of G and we think of

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1983

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