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Uniformly Quasiregular Maps on the Compactified Heisenberg Group

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Abstract

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification \(\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}\) of the Heisenberg group ℍ1 equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere \({\mathbb{S}}^{n} \) was proven by Martin (Conform. Geom. Dyn. 1:24–27, 1997). Moreover, we construct uniformly quasiregular mappings on \(\bar{ {\mathbb{H}}}^{1}\) with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on \(\bar{ {\mathbb{H}}}^{1}\) there exists a measurable CR structure μ which is equivariant under the semigroup Γ generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.

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Authors and Affiliations

  1. Department of Mathematics, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Zoltán M. Balogh & Katrin Fässler

  2. Department of Mathematics, Aalto University, P.O. Box 11100, 00076, Aalto, Finland

    Kirsi Peltonen

Authors
  1. Zoltán M. Balogh
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  2. Katrin Fässler
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  3. Kirsi Peltonen
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Correspondence to Katrin Fässler.

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Communicated by Fulvio Ricci.

The authors were supported by Swiss National Science Foundation, European Research Council Project GALA and European Science Foundation Project HCAA.

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Balogh, Z.M., Fässler, K. & Peltonen, K. Uniformly Quasiregular Maps on the Compactified Heisenberg Group. J Geom Anal 22, 633–665 (2012). https://doi.org/10.1007/s12220-010-9205-5

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