Local rigidity for hyperbolic groups with Sierpi\'nski carpet boundaries

Let $G$ and $\tilde{G}$ be Kleinian groups whose limit sets $S$ and $\tilde S$, respectively, are homeomorphic to the standard Sierpi'nski carpet, and such that every complementary component of each of $S$ and $\tilde{S}$ is a round disc. We assume that the groups $G$ and $\tilde{G}$ 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 $S$ to $\tilde{S}$ is the restriction of a M"obius transformation that takes $S$ onto $\tilde{S}$, 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 $G$ is a torsion-free hyperbolic group whose boundary at infinity ${\text{dee}}_{\mathrm{\infty }}G$ is a Sierpi'nski carpet that embeds quasisymmetrically into the standard 2-sphere. Then there exists a group $H$ that contains $G$ as a finite index subgroup and such that any quasisymmetric map $f$ between open connected subsets of ${\text{dee}}_{\mathrm{\infty }}G$ is the restriction of the induced boundary map of an element $h\in H$.