36 research outputs found
Local rigidity for hyperbolic groups with Sierpi\'nski carpet boundaries
Let and be Kleinian groups whose limit sets and , respectively, are homeomorphic to the standard Sierpi\'nski carpet, and
such that every complementary component of each of and is a
round disc. We assume that the groups and act cocompactly on
triples on their respective limit sets. The main theorem of the paper states
that any quasiregular map (in a suitably defined sense) from an open connected
subset of to is the restriction of a M\"obius transformation
that takes onto , in particular it has no branching. This theorem
applies to the fundamental groups of compact hyperbolic 3-manifolds with
non-empty totally geodesic boundaries.
One consequence of the main theorem is the following result. Assume that
is a torsion-free hyperbolic group whose boundary at infinity \dee_\infty G
is a Sierpi\'nski carpet that embeds quasisymmetrically into the standard
2-sphere. Then there exists a group that contains as a finite index
subgroup and such that any quasisymmetric map between open connected
subsets of \dee_\infty G is the restriction of the induced boundary map of an
element .Comment: 14 page