Rigidity of Schottky sets

TL;DR

This paper proves that Schottky sets of measure zero are rigid under quasisymmetric maps, being only transformable by Möbius transformations, while those of positive measure admit non-trivial quasisymmetric maps, with applications to hyperbolic geometry.

Contribution

It establishes rigidity results for Schottky sets of measure zero and demonstrates the existence of non-trivial maps for positive measure sets, advancing understanding of geometric structures.

Findings

Measure zero Schottky sets are quasisymmetrically rigid.

Positive measure Schottky sets admit non-trivial quasisymmetric maps.

Applications to convex hyperbolic subsets with geodesic boundaries.

Abstract

We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.