Measurable Rigidity for Kleinian groups

2014/06/18 by Jeon, Woojin, Ohshika, Ken'ichi
#30F40 #37A40 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1406.4594

Abstract

Let G, H be two Kleinian groups with homeomorphic quotients \mathbb H³/G and \mathbb H³/H. We assume that G is of divergence type, and consider the Patterson-Sullivan measures of G and H. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map \widehat k from the limit set Λ_G of G to that of H is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such \widehat k always exists. In fact, we shall construct \widehat k concretely from the Cannon-Thurston maps of G and H.

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