Rigidity of certain solvable actions on the sphere

TL;DR

This paper studies the conformal actions of Baumslag-Solitar groups on spheres, showing they are not locally rigid but do preserve a smooth conformal structure under perturbations.

Contribution

It introduces a weak form of local rigidity for certain solvable group actions on spheres, expanding understanding of conformal group actions.

Findings

Actions are not locally rigid in higher dimensions.

Perturbations preserve a smooth conformal structure.

Provides insight into rigidity phenomena for solvable groups.

Abstract

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure.