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Quasiconvexity in the Heisenberg group

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Abstract

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.

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Acknowledgements

Research for this paper was conducted during visits of various subsets of the authors to the University of Illinois and the University of Cincinnati. The hospitality of these institutions is appreciated.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, French Hall West, Cincinnati, OH, 45221-0025, USA

    David A. Herron

  2. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Anton Lukyanenko

  3. Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL, 61801, USA

    Jeremy T. Tyson

Authors
  1. David A. Herron
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  2. Anton Lukyanenko
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  3. Jeremy T. Tyson
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Corresponding author

Correspondence to Anton Lukyanenko.

Additional information

Dedicated to William Goldman on the occasion of his 60th birthday.

J. T. T. was supported by Simons Foundation Collaboration Grant 353627 ‘Geometric Analysis in Sub-Riemannian and Metric Spaces’. D.A.H. was supported by the Charles Phelps Taft Research Center. A.L. was supported by NSF RTG Grant DMS-1045119. All three authors were supported by the NSF Grant DMS-1500454.

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Herron, D.A., Lukyanenko, A. & Tyson, J.T. Quasiconvexity in the Heisenberg group. Geom Dedicata 192, 157–170 (2018). https://doi.org/10.1007/s10711-017-0257-6

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  • DOI: https://doi.org/10.1007/s10711-017-0257-6

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