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We study the closed group of homeomorphisms of the boundary of realhyperbolic space generated by a cocompact Kleinian group $G_1$ and aquasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show thatif the conjugacy $h$ is not conformal then this group contains a non-trivialone parameter subgroup. This leads to rigidity results; for example, Mostowrigidity is an immediate consequence. We are also able to prove a relativeversion of Mostow rigidity, called pattern rigidity. For a cocompact group $G$,by a $G$-invariant pattern we mean a $G$-invariant collection of closed propersubsets of the boundary of hyperbolic space which is discrete in the space ofcompact subsets minus singletons. Such a pattern arises for example as thecollection of translates of limit sets of finitely many infinite indexquasiconvex subgroups of $G$. We prove that (in dimension at least three) for$G_1, G_2$ cocompact Kleinian groups, any quasiconformal map pairing a$G_1$-invariant pattern to a $G_2$-invariant pattern must be conformal. Thisgeneralizes a previous result of Schwartz who proved rigidity in the case oflimit sets of cyclic subgroups, and Biswas-Mj who proved rigidity for PoincareDuality subgroups.

Flows, fixed points and rigidity for Kleinian groups