2009/03/13 by Kingshook Biswas, Biswas, Kingshook
Mathematics · #57M50 #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.0903.2419
openalex publication_date 2009/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group G1 and a quasiconformal conjugate h-1G2 h of a cocompact group G2. We show that if the conjugacy h is not conformal then this group contains a non-trivial one parameter subgroup. This leads to rigidity results; for example, Mostow rigidity is an immediate consequence. We are also able to prove a relative version of Mostow rigidity, called pattern rigidity. For a cocompact group G, by a G-invariant pattern we mean a G-invariant collection of closed proper subsets of the boundary of hyperbolic space which is discrete in the space of compact subsets minus singletons. Such a pattern arises for example as the collection of translates of limit sets of finitely many infinite index quasiconvex subgroups of G. We prove that (in dimension at least three) for G1, G2 cocompact Kleinian groups, any quasiconformal map pairing a G1-invariant pattern to a G2-invariant pattern must be conformal. This generalizes a previous result of Schwartz who proved rigidity in the case of limit sets of cyclic subgroups, and Biswas-Mj who proved rigidity for Poincare Duality subgroups.