We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet S 3 is a Euclidean isometry. For carpets in a more general family, the standard 1/p-Sierpiński carpets S p , p ≥ 3 odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that S p and S q are quasisymmetrically equivalent only if p = q. The main tool in the proof for these facts is a new invariant-a certain discrete modulus of a path family-that is preserved under quasisymmetric maps of carpets.

Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasi-isometrically co-Hopfian Gromov hyperbolic space with a Sierpiński carpet boundary at infinity. In addition, we give a complete description of the quasisymmetry group of the constructed Sierpiński carpet. This group is uncountable and coincides with the group of bi-Lipschitz transformations.

We prove that if ξ is a quasisymmetric homeomorphism between Sierpiński carpets that are the Julia sets of postcritically-finite rational maps, then ξ is the restriction of a Möbius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.Theorem 1.10. Let f : C → C be a subhyperbolic rational map whose Julia set J (f ) is a Sierpiński carpet. Then the peripheral circles of the Sierpiński carpet J (f ) are uniform quasicircles, they are uniformly relatively separated, and they occur on all locations and scales. Moreover, J (f ) is a porous set, and in particular, has measure zero.See Sections 2 and 5 for an explanation of the terminology and for the proof.

We study a quasisymmetric version of the classical Koebe uniformization theorem in the context of Ahlfors regular metric surfaces. In particular, we prove that an Ahlfors 2-regular metric surface X homeomorphic to a finitely connected domain in the standard 2-sphere S 2 is quasisymmetrically equivalent to a circle domain in S 2 if and only if X is linearly locally connected and its completion is compact. We also give a counterexample in the countably connected case.

Abstract. A relative Schottky set in a planar domain Ω is a subset of Ω obtained by removing from Ω disjoint open geometric discs. In this paper we study quasisymmetric and related maps between relative Schottky sets of zero measure. We prove, in particular, that under mild geometric assumptions quasisymmetric maps between such sets in Jordan domains are conformal and locally biLipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish a local quasisymmetric rigidity for relative Schottky sets in the unit disc.