Conformal and CR mappings on Carnot groups

Abstract

We consider a class of stratified groups with a CR structure and a compatible control distance. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component. As examples, we consider free groups on two generators, and show that these admit very simple polynomial embeddings in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {C}^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that induce their CR structure.

Locations

• Proceedings of the American Mathematical Society Series B - View - PDF
• UNSWorks (University of New South Wales, Sydney, Australia) - View - PDF