Conrad Plaut

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.

Metric Spaces of Curvature ⩾ k

This article is intended to be an introduction to the theory of metric spaces of curvature bounded below, and a survey of recent results. Time and space constraints have prevented it from being as comprehensive as we originally had planned, but we hope that this article will provide a good beginning point for a student or mathematician interested in this area, and a common reference for future papers on the subject. Through Section 9 we have tried to present good enough sketches of nearly all arguments, so that a dedicated reader can fill in the remaining details without too much difficulty. In Section 9 we provide fewer details, and in the last sections we mostly survey known results.

Homogeneous spaces of curvature bounded below

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces.

Covering group theory for topological groups

Abstract We develop a covering group theory for a large category of “coverable” topological groups, with a generalized notion of “cover”. Coverable groups include, for example, all metrizable, connected, locally connected groups, and even many totally disconnected groups. Our covering group theory produces a categorial notion of fundamental group, which, in contrast to traditional theory, is naturally a (prodiscrete) topological group. Central to our work is a link between the fundamental group and global extension properties of local group homomorphisms. We provide methods for computing the fundamental group of inverse limits and dense subgroups or completions of coverable groups. Our theory includes as special cases the traditional theory of Poincare, as well as alternative theories due to Chevalley, Tits, and Hoffmann–Morris. We include a number of examples and open problems.

Spaces of Wald-Berestovskii curvature bounded below

We consider inner metric spaces of curvature bounded below in the sense of Wald, without assuming local compactness or existence of minimal curves. We first extend the Hopf-Rinow theorem by proving existence, uniqueness, and “almost extendability” of minimal curves from any point to a denseGδ subset. An immediate consequence is that Alexandrov’s comparisons are meaningful in this setting. We then prove Toponogov’s theorem in this generality, and a rigidity theorem which characterizes spheres. Finally, we use our characterization to show the existence of spheres in the space of directions at points in a denseGδ set. This allows us to define a notion of “local dimension” of the space using the dimension of such spheres. If the local dimension is finite, the space is an Alexandrov space.

Covering group theory for locally compact groups

Abstract In an earlier paper we introduced a covering group theory for a category of “coverable” topological groups, including a generalized notion of universal cover. In this paper we characterize coverable locally compact groups. As an application we show that the classical covering group theories of Poincare and Chevalley, as well as a variants due to Tits and Hofmann–Morris, are all equivalent for locally compact groups, and are strictly special cases of our theory (which does not require any form of local simple connectivity). As a second application we show the existence of an inverse sequence of locally compact groups, whose bonding homomorphisms are open surjections with discrete kernel, such that the natural projections from the inverse limit are not surjective.

Geometrizing Infinite Dimensional Locally Compact Groups

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.

Covering group theory for compact groups

We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism � : 2