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Large scale geometry of negatively curved R n o R
We classify all negatively curved R n oR up to quasiisometry. We show that all quasiisometries between such manifolds (except when they are biLipschitz to the real hyperbolic spaces) are almostExpand
Quasi-isometric rigidity of Fuchsian buildings
Abstract We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism betweenExpand
Growth of relatively hyperbolic groups
We show that a finitely generated group that is hyperbolic relative to a collection of proper subgroups either is virtually cyclic or has uniform exponential growth.
Large scale geometry of negatively curved $\R^n \rtimes \R$
We classify all negatively curved $\R^n \rtimes \R$ up to quasiisometry. We show that all quasiisometries between such manifolds (except when they are biLipschitz to the real hyperbolic spaces) areExpand
The Tits boundary of a CAT(0) 2-complex
We investigate the Tits boundary of CAT(0) 2-complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the TitsExpand
Metric space inversions, quasihyperbolic distance, and uniform spaces
We define a notion of inversion valid in the general metric space setting. We establish several basic facts concerning inversions; e.g., they are quasimobius homeomorphisms and quasihyperbolicallyExpand
Quasisymmetric maps on the boundary of a negatively curved solvable Lie group
We describe all the self quasisymmetric maps on the ideal boundary of a particular negatively curved solvable Lie group. As applications, we derive some rigidity properties for quasiisometries of theExpand
Quasiconformal maps on non-rigid Carnot groups
We study quasiconformal maps on non-rigid Carnot groups equipped with Carnot metric. We show that for most non-rigid Carnot groups N, all quasiconformal maps on N must be biLipschitz.
Uniformity from Gromov hyperbolicity
We show that in a metric space $X$ with annular convexity, uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the Gromov boundary agrees with that on theExpand
Rigidity of quasiconformal maps on Carnot groups
  • Xiangdong Xie
  • Mathematics
  • Mathematical Proceedings of the Cambridge…
  • 14 August 2013
Abstract We show that quasiconformal maps on many Carnot groups must be biLipschitz. In particular, this is the case for 2-step Carnot groups with reducible first layer. These results haveExpand
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