Quasiconformal mappings on the Heisenberg group: An overview

TL;DR

This paper provides an overview of quasiconformal mappings on the Heisenberg group, highlighting key similarities and differences with classical theory, and discusses extensions, known results, and open problems in the field.

Contribution

It offers a comprehensive overview of the Korányi-Reimann theory on the Heisenberg group, emphasizing analogies, differences, and extensions beyond classical settings.

Findings

Comparison of Heisenberg group and classical quasiconformal theory

Discussion of extensions to more general spaces

Summary of known results and open problems

Abstract

We present a brief overview of the Kor\'anyi-Reimann theory of quasiconformal mappings on the Heisenberg group stressing on the analogies as well as on the differences between the Heisenberg group case and the classical two-dimensional case. We examine the extensions of the theory to more general spaces and we state some known results and open problems.