This is an intersting exposition of some recent topics in quasiconformal geometry in Euclidean spaces and other metric spaces, in particular fractal 2-spheres and Sierpiński carpets.
The author starts by recalling the notion of quasiconformal maps and other related notions such as quasisymmetric and quasi-Möbius maps Ahlfors Q-regular spaces and Q-Loewner space. He then discusses the “quasi-symmetric uniformization problem”, that is, given a metric space \(X\) which is homeomorphic to some “standard” metric space \(Y\), to find conditions under which \(X\) is quasisymmetrically equivalent to \(Y\). The author mentions results of Tukia and Väisälä, valid in the case where \(X\) is homeomorphic to a circle, as well as work of S. Semmes on the quasisymmetric characterization of spheres of dimension \(\geq 3\). He then mentions joint work of himself and B. Kleiner in the case where \(X\) is homeomorphic to the 2-sphere.
The author then discusses quasisymmetric maps on boundaries of Gromov hyperbolic spaces which are induced by quasi-isometries between these spaces. This leads him to a discussion of Cannon’s conjecture stating that if \(G\) is a Gromov hyperbolic group whose boundary is homeomorphic to the 2-sphere, then there exists an isometric properly discontinuous and cocompact action of \(G\) on hyperbolic 3-space. The author presents then several recent results of himself and Kleiner related to that conjecture. For instance, they prove that if the Ahlfors regular conformal dimension of such a group \(G\) is attained as a minimum, then this boundary is quasisymmetrically equivalent to the standard 2-sphere.
After the discussion of Gromov hyperbolic spaces, the author discusses problems of quasiconformal geometry related to the theory of post-critically finite rational maps of the Riemann sphere as developed by Thurston and Douady-Hubbard. He presents work by D. Meyer and recent joint work by himself and Meyer on that subject giving a characterization of maps of the sphere conjugate to rational maps in terms of quasisymmetric geometry.
In the last part of the paper, the author discusses the quasiconformal geometry of Sierpiński carpets. He describes a recent result by himself, B. Kleiner and S. Merenkov stating that a round carpet in the two sphere (i.e. a carpet whose peripheral circles are round circles) is quasisymmetrically rigid if and only if it has measure zero. He also discusses a rigidity conjecture by Kapovich and Kleiner on hyperbolic groups whose boundary is homeomorphic to the standard Sierpiński carpet, and a recent resut by himself and Merenkov on quasisymmetric rigidity of square carpets (i. e. carpets whose peripheral circles are boundaries of geometric squares).
For the entire collection see [Zbl 1095.00005].