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28
Conformal dimension and Gromov hyperbolic groups with 2sphere boundary
, 2003
"... Suppose G is a Gromov hyperbolic group, and ∂∞G is quasisymmetrically homeomorphic to an Ahlfors Qregular metric 2sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H³. ..."
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Cited by 16 (3 self)
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Suppose G is a Gromov hyperbolic group, and ∂∞G is quasisymmetrically homeomorphic to an Ahlfors Qregular metric 2sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H³.
The quasiconformal Jacobian problem
 In In the tradition of Ahlfors and Bers, III
"... Which nonnegative functions can arise, up to a bounded multiplicative error, as Jacobian determinants Jf(x) = det(Df(x)) of quasiconformal mappings f: R n → R n, n ≥ 2? Which metric spaces are biLipschitz equivalent to R n, n ≥ 2? We first survey what is known about these problems, and point out h ..."
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Cited by 15 (3 self)
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Which nonnegative functions can arise, up to a bounded multiplicative error, as Jacobian determinants Jf(x) = det(Df(x)) of quasiconformal mappings f: R n → R n, n ≥ 2? Which metric spaces are biLipschitz equivalent to R n, n ≥ 2? We first survey what is known about these problems, and point out how it follows from recent work of B. Kleiner
NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
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Cited by 12 (0 self)
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Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
Mappings with convex potentials and the quasiconformal Jacobian problem
 ILLINOIS J. MATH
, 2005
"... This paper concerns convex functions that arise as potentials of quasiconformal mappings. Several equivalent definitions for such functions are given. We use them to construct quasiconformal mappings whose Jacobian determinants are singular on a prescribed set of Hausdorff dimension less than 1. ..."
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Cited by 8 (6 self)
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This paper concerns convex functions that arise as potentials of quasiconformal mappings. Several equivalent definitions for such functions are given. We use them to construct quasiconformal mappings whose Jacobian determinants are singular on a prescribed set of Hausdorff dimension less than 1.
The branch set of a quasiregular mapping
 in: Proceedings of the ICM Beijing, Higher Education
, 2002
"... We discuss the issue of branching in quasiregular mapping, and in particular the relation between branching and the problem of finding geometric parametrizations for topological manifolds. Other recent progress and open problems of a more function theoretic nature are also presented. ..."
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Cited by 8 (2 self)
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We discuss the issue of branching in quasiregular mapping, and in particular the relation between branching and the problem of finding geometric parametrizations for topological manifolds. Other recent progress and open problems of a more function theoretic nature are also presented.
The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity
 ICM Proceedings
, 2006
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Rapidly growing entire functions with three singular values
, 2007
"... We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic dominates an arbitrarily prescribed function. ..."
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Cited by 5 (0 self)
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We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic dominates an arbitrarily prescribed function.
Quasisymmetric rigidity of Sierpiński carpets
, 2006
"... We prove that any quasisymmetric selfmap of the standard Sierpiński carpet is a rotation or a reflection. For a more general family of carpets, so called standard Sierpiński carpets, we show that the group of quasisymmetric selfmaps is finite. We also show that any two distinct standard Sierpiński ..."
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Cited by 3 (1 self)
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We prove that any quasisymmetric selfmap of the standard Sierpiński carpet is a rotation or a reflection. For a more general family of carpets, so called standard Sierpiński carpets, we show that the group of quasisymmetric selfmaps is finite. We also show that any two distinct standard Sierpiński carpets are not quasisymmetric to each other. The main tool is a new invariant for quasisymmetric maps of carpets, a modulus of a curve family with respect to a carpet. 1
A SIERPIŃSKI CARPET WITH THE COHOPFIAN PROPERTY
"... Abstract. Motivated by questions in geometric group theory we define a quasisymmetric coHopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasiisometrically coHopfian Gromov hyperbolic space with a Sierpiński ..."
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Cited by 3 (1 self)
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Abstract. Motivated by questions in geometric group theory we define a quasisymmetric coHopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasiisometrically coHopfian 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 biLipschitz transformations. 1.