Rigidity of schottky sets

Mario Bonk, Bruce Kleiner, Sergei Merenkov

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)409-443
Number of pages35
JournalAmerican Journal of Mathematics
Volume131
Issue number2
DOIs
StatePublished - 2009

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Rigidity
Totally Geodesic
Hyperbolic Space
Homeomorphism
Unit Sphere
Disjoint
Ball
Union
Complement
Restriction
Subset
Zero

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Rigidity of schottky sets. / Bonk, Mario; Kleiner, Bruce; Merenkov, Sergei.

In: American Journal of Mathematics, Vol. 131, No. 2, 2009, p. 409-443.

Research output: Contribution to journalArticle

Bonk, Mario ; Kleiner, Bruce ; Merenkov, Sergei. / Rigidity of schottky sets. In: American Journal of Mathematics. 2009 ; Vol. 131, No. 2. pp. 409-443.
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