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Quasisymmetric embeddings of a closed ball inextensible in neighbourhoods of any boundary points. (English) Zbl 0747.30016

For a set \(X\) in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), an embedding \(f\colon X\to\mathbb{R}^n\) is called quasisymmetric if there is a homeomorphism \(c\colon [0,\infty)\to [0,\infty)\) such that \(| f(y)- f(x)|\le c(r)| f(z)-f(x)|\) for all \(x,y,z\in X\) with \(| y-x|\leq r| z-x|\). In the case \(n=2\), it is well known that every quasisymmetric embedding of a closed disc \(\overline{B^2}\) into \(\mathbb{R}^2\) can be extended to a quasiconformal automorphism of \(\mathbb{R}^2\).
On the other hand, F. W. Gehring [Tr. Mezhdunarod. Kongr. Mat., Moskva 1966, 313–318 (1968; Zbl 0193.03803)] proved that there are quasisymmetric embeddings of a closed ball \(B^3\) into \(\mathbb{R}^3\) which cannot be extended to embeddings of an open neighborhood \(U\) of \(B^3\).
In this paper, the author constructs a quasisymmetric embedding of a closed ball \(B\) into \(\mathbb{R}^3\) which is quasiconformal inside \(B\) and cannot be extended to an embedding of any neighborhood of any boundary point of \(B\). In his argument he constructs a geometrically finite Kleinian group acting on \(\mathbb{R}^3\) whose limit set is a wildly knotted sphere and uses an ingenious construction of the spherical covering.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)

Citations:

Zbl 0193.03803
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