Quasiregular maps S3 → S3 with wild branch sets

Two examples of quasiregular maps S3 → S3 that branch on a wild Cantor set are constructed. As an application it is shown that certain interesting 3-dimensional metric spaces recently constructed by Semmes admit Lipschitz branched covers onto S3. Moreover, it is shown that a uniformly quasiconformal group of Freedman and Skora acting on S3 and not topologically conjugate to a Möbius group is quasiregularly semiconjugate to a Möbius group.