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

TL;DR

This paper proves a local rigidity theorem for hyperbolic groups with Sierpiński carpet boundaries, showing that certain quasiregular maps are Möbius transformations, with implications for group actions and boundary maps.

Contribution

It establishes a local rigidity result for hyperbolic groups with Sierpiński carpet boundaries, characterizing quasiregular maps as Möbius transformations and extending to boundary group actions.

Findings

Quasiregular maps between limit sets are Möbius transformations.

Rigidity applies to hyperbolic 3-manifold groups with geodesic boundaries.

Boundary maps are induced by group elements in a finite index supergroup.

Abstract

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 $\dee_\infty G$ 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 $\dee_\infty G$ is the restriction of the induced boundary map of an element $h\in H$.