Quasisymmetric embeddings in Euclidean spaces

Abstract:

We consider quasi-symmetric embeddings $f:G \to {R^n}$, $G$ open in ${R^p}$, $p \leqslant n$. If $p = n$, quasi-symmetry implies quasi-conformality. The converse is true if $G$ has a sufficiently smooth boundary. If $p < n$, the Hausdorff dimension of $fG$ is less than $n$. If $fG$ has a finite $p$-measure, $f$ preserves the property of being of $p$-measure zero. If $p < n$ and $n \geqslant 3$, ${R^n}$ contains a quasi-symmetric $p$-cell which is topologically wild. We also prove auxiliary results on the relations between Hausdorff measure and Čech cohomology.