  We study the closed group of homeomorphisms of the boundary of real
hyperbolic space generated by a cocompact Kleinian group $G_1$ and a
quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. 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
$G_1, G_2$ cocompact Kleinian groups, any quasiconformal map pairing a
$G_1$-invariant pattern to a $G_2$-invariant pattern must be conformal. 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.
