Xiangdong Xie

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.

Geometrically Finite Complex Hyperbolic Manifolds

The aim of this paper is to study geometry and topology of geometrically finite complex hyperbolic manifolds, especially their ends, as well as geometry of their holonomy groups. This study is based on our structural theorem for discrete groups acting on Heisenberg groups, on the fiber bundle structure of Heisenberg manifolds, and on the existence of finite coverings of a geometrically finite manifold such that their parabolic ends have either Abelian or 2-step nilpotent holonomy. We also study an interplay between Kahler geometry of complex hyperbolic n-manifolds and Cauchy–Riemannian geometry of their boundary (2n-1)-manifolds at infinity, and this study is based on homotopy equivalence of manifolds and isomorphism of fundamental groups.

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.

Discrete actions on nilpotent Lie groups and negatively curved spaces

Abstract The aim of this paper is to study dynamics of a discrete isometry group action in a pinched Hadamard manifold nearby its parabolic fixed points. Due to Margulis Lemma, such an action on corresponding horospheres is virtually nilpotent, so we solve the problem by establishing a structural theorem for discrete groups acting on connected nilpotent Lie groups. As applications, we show that parabolic fixed points of a discrete isometry group cannot be conical limit points, that the fundamental groups of geometrically finite orbifolds with pinched negative sectional curvature are finitely presented, and the orbifolds themselves are topologically finite.

Tits alternative for closed real analytic 4-manifolds of nonpositive curvature

Abstract 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.

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.

The Tits boundary of a (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 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.

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.

Foldable cubical complexes of nonpositive curvature

We study finite foldable cubical complexes of nonpositive cur- vature (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. AMS Classification 20F65, 20F67; 53C20