Some examples of dynamically defined ambient homogeneous wild knots in higher dimensions

TL;DR

This paper demonstrates that certain wild sphere limit sets of Kleinian groups acting conformally on higher-dimensional spheres are ambient homogeneous, meaning they can be mapped onto themselves with homeomorphisms that send any point to any other.

Contribution

It proves that specific wild sphere limit sets in higher dimensions are ambient homogeneous, extending understanding of their symmetry properties.

Findings

Wild sphere limit sets are ambient homogeneous in dimensions 3 to 7.

Homeomorphisms exist that map any point of the wild sphere to any other.

The result applies to limit sets constructed in prior work by the authors.

Abstract

In this paper we consider the Kleinian groups acting conformally on the sphere $\mathbb{S}^{n+2}$ $(1\leq{n}\leq5)$ which have as limit sets wild spheres $K^n$ which were constructed in \cite{BHV} and prove that $K^n$ is ambient homogeneous. In other words, given two points $p,\,\,q\in{K}$ there exists a homeomorphism $\psi:\mathbb{S}^{n+2}\rightarrow\mathbb{S}^{n+2} $ such that $\psi(K)=K$ and $\psi(p)=q$.