Urs Lang

Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

We discuss a variation of Gromovs notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, euclidean buildings, and homogeneous or pinched negatively curved Hadamard manifolds. Among others, we prove a quasisymmetric embedding theorem for spaces with finite Nagata dimension in the spirit of theorems of Assouad and Dranishnikov, and we characterize absolute Lipschitz retracts of finite Nagata dimension.

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.

Extensions of Lipschitz maps into Hadamard spaces

Abstract. We prove that every

INJECTIVE HULLS OF CERTAIN DISCRETE METRIC SPACES AND GROUPS

\lambda

Extendability of Large-Scale Lipschitz Maps

-Lipschitz map

Jung's Theorem for Alexandrov Spaces of Curvature Bounded Above

f : S \to Y

THE EXISTENCE OF COMPLETE MINIMIZING HYPERSURFACES IN HYPERBOLIC MANIFOLDS

defined on a subset of an arbitrary metric space X possesses a

Convex Hulls in Singular Spaces of Negative Curvature

c \lambda

Quasiflats in Hadamard spaces

-Lipschitz extension

Quasi-minimizing surfaces in hyperbolic space

\bar{f} : X \to Y