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.
math.CV↗
arXiv subjects
Xiangdong Xie
Publications and source records attributed to Xiangdong Xie.
Rigidity of quasiconformal maps on Carnot groups
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 have implications for the rigidity of quasiisometries between negatively curved solvable Lie groups.
math.CV↗
Quasiisometries of negatively curved homogeneous manifolds associated with Heisenberg groups
We study quasiisometries between negatively curved homogeneous manifolds associated with diagonalizable derivations on Heisenberg algebras. We classify these manifolds up to quasiisometry, and show that all quasiisometries between such manifolds (except when they are complex hyperbolic spaces) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.
math.GR↗
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 prove a Liouville type theorem, and derive some rigidity properties for quasiisometries of the solvable Lie group.
math.GR↗
A Rigidity Property of Some Negatively Curved Solvable Lie Groups
We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with respect to the visual metric. We also define parabolic visual metrics on the ideal boundary of Gromov hyperbolic spaces and relate them to visual metrics.
math.GR↗
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) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.
math.GR↗
Quasiisometries between negatively curved Hadamard manifolds
Let X, Y be the universal covers of two compact Riemannian manifolds (with dimension not equal to 4) with negative sectional curvature. Then every quasiisometry between them lies at a finite distance from a bilipschitz homeomorphism.
math.GR↗
Quasimobius maps preserve uniform domains
We show that if a domain D in a geodesic metric space is quasimobius to a uniform domain in some metric space, then D is also uniform.
math.CV↗
Growth of relatively hyperbolic groups
We show that a relatively hyperbolic group either is virtually cyclic or has uniform exponential growth.
math.GR↗
Foldable cubical complexes of nonpositive curvature
We study finite foldable cubical complexes of nonpositive curvature (in the sense of A.D. Alexandrov). We show that such a complex X admits a graph of spaces decomposition. It is also shown that when dim X=3, X contains a closed rank one geodesic in the 1-skeleton unless the universal cover of X is isometric to the product of two CAT(0) cubical complexes.
math.MG↗
Mostow rigidity for Fuchsian buildings
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 between the Fuchsian buildings. It follows that Mostow rigidity holds for Fuchsian buildings: if a group acts properly and cocompactly on two Fuchsian buildings X and Y, then X and Y are equivariantly isomorphic.
math.GR↗
Tits alternative for closed real analytic 4-manifolds of nonpositive curvature
We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theory of general metric spaces of nonpositive curvature.
math.GR↗
Tits boundary of CAT(0) 2-complexes
We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two CAT(0) 2-complexes are quasi-isometric then the cores of their Tits boundaries are bi-Lipschitz.
math.GR↗
Groups acting on CAT(0) square complexes
We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or contains a free group of rank two. In addition we discuss when a group generated by two hyperbolic isometries contains a free group of rank two and when two points in the ideal boundary of a CAT(0) 2-complex at Tits distance $π$ apart are the endpoints of a geodesic in the 2-complex.
math.GR↗