Quasiconformality and hyperbolic skew

Abstract

We prove that if $f:{\mathbb{B}}^n\to {\mathbb{B}}^n$, for $n\ge 2$, is a homeomorphism with bounded skew over all equilateral hyperbolic triangles, then $f$ is in fact quasiconformal. Conversely, we show that if $f:{\mathbb{B}}^n\to {\mathbb{B}}^n$ is quasiconformal then $f$ is $\eta$-quasisymmetric in the hyperbolic metric, where $\eta$ depends only on $n$ and $K$. We obtain the same result for hyperbolic $n$-manifolds. Analogous results in ${\mathbb{R}}^n$, and metric spaces that behave like ${\mathbb{R}}^n$, are known, but as far as we are aware, these are the first such results in the hyperbolic setting, which is the natural metric to use on ${\mathbb{B}}^n$.

References in corpus (1)

• Superattracting fixed points of quasiregular mappings