Quasiconformal maps on model Filiform groups

TL;DR

This paper characterizes all quasiconformal maps on higher-dimensional model Filiform groups with Carnot metrics, revealing their special forms and implications for the geometry of certain Lie groups.

Contribution

It provides a complete description of quasiconformal maps on model Filiform groups, including non-smooth maps, and explores their geometric properties and implications.

Findings

All quasiconformal maps are biLipschitz and preserve multiple foliations.

Maps have very special forms, including non-smooth ones.

Results impact understanding of large scale geometry of nilpotent and solvable Lie groups.

Abstract

We describe all quasiconformal maps on the higher (real and complex) model Filiform groups equipped with the Carnot metric, including non-smooth ones. These maps have very special forms. In particular, they are all biLipschitz and preserve multiple foliations. The results in this paper have implications to the large scale geometry of nilpotent Lie groups and negatively curved solvable Lie groups.