Abstract
We call the 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 Möbius transformation on S n. In the other direction we show that every Schottky set in S 2 of positive measure admits nontrivial 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.
Original language | English (US) |
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Pages (from-to) | 409-443 |
Number of pages | 35 |
Journal | American Journal of Mathematics |
Volume | 131 |
Issue number | 2 |
DOIs | |
State | Published - 2009 |
ASJC Scopus subject areas
- General Mathematics