We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called "detour sets", which resemble the Sierpiński gasket.The main theorem is that if K ⊂ R n is a detour set and its complementary components are sufficiently regular, then K is W 1,p -removable for p > n. Several examples and constructions of sets where the theorem applies are given, including the Sierpiński gasket, Apollonian gaskets, and Julia sets.

A circle domain Ω in the Riemann sphere is conformally rigid if every conformal map from Ω onto another circle domain is the restriction of a Möbius transformation. We show that circle domains satisfying a certain quasihyperbolic condition, which was considered by Jones and Smirnov [11], are conformally rigid. In particular, Hölder circle domains and John circle domains are all conformally rigid. This provides new evidence for a conjecture of He and Schramm relating rigidity and conformal removability.

We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop [Bi15]. The proof involves a new technique of constructing an exceptional homeomorphism from R 2 into some non-planar surface S, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem [BK02]. We also prove that all homeomorphic copies of the Sierpiński gasket are non-removable for continuous Sobolev functions of the class W 1,p for 1 ≤ p ≤ 2, thus complementing and sharpening the results of the author's previous work [Nt17].

We provide a David extension result for circle homeomorphisms conjugating two dynamical systems such that parabolic periodic points go to parabolic periodic points, but hyperbolic points can go to parabolics as well. We use this result, in particular, to produce matings of anti-polynomials and necklace reflection groups, show conformal removability of the Julia sets of geometrically finite polynomials and of the limit sets of necklace reflection groups, and give a new proof of the existence of Suffridge polynomials (extremal points in certain spaces of univalent maps).

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