Quasiconvexity in the Heisenberg group

David A. Herron, Anton Lukyanenko, Jeremy T. Tyson

Research output: Contribution to journalArticlepeer-review


We show that if A is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of A is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff 3-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.

Original languageEnglish (US)
Pages (from-to)157-170
Number of pages14
JournalGeometriae Dedicata
Issue number1
StatePublished - Feb 1 2018


  • Carnot–Caratheodory
  • Heisenberg group
  • Quasiconvexity

ASJC Scopus subject areas

  • Geometry and Topology


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