CAT(κ)-Spaces: Construction and Concentration

Abstract

Basic constructions of spaces X with K ≤ 0 are described. Bibliography: 14 titles. To Victor Abramovich Zalgaller We expose spaces X with negative curvature, having in mind applications to fractally hyperbolic groups, such as random groups and infinite Burnside groups. Originally, these spaces were introduced by Aleksandrov in the axiomatization spirit, and a similar class (of convex spaces) was later isolated by Busemann. Till relatively recently, the major thrust of geometric research was laid on suppressing singularities, emphasizing the properties equally shared by smooth and singular spaces, and proving regularization theorems claiming, under certain assumptions, that X can be approximated by smooth manifolds with curvature K ≤ 0. This was accomplished for surfaces in a famous treatise by Aleksandrov and Zalgaller. However, the bulk of spaces with K ≤ 0 is badly singular, starting from trees and most abundant among 2-polyhedra. Furthermore, almost all "natural" spaces with K ≤ 0, such as the Bruhat-Tits buildings, are nonsmooth and (unlike trees) usually cannot be approximated by smooth spaces. But geometers remained unaware of this for a stretch of time. From another angle, the idea of negative curvature was injected into the group theory by Dehn and grew up into the small cancellation theory. In the course of the development, the geometric roots were forgotten, and the role of curvature was reduced to a metaphor. (Algebraists do not trust geometry.) It eventually turned out that the geometric language of Dehn and Aleksandrov (sometimes slightly modified and/or generalized) accomplishes many needs of combinatorial group theory more efficiently than the combina-torial language. Summing up, geometry furnishes a proper language, while the combinatorial group theory (especially random groups) provides a pool of objects for a meaningful usage of this language. In this paper, we present basic constructions of spaces X with K ≤ 0 relevant for applications in group theory (see [8]), as well as basic isoperimetric concentration properties of maps of metric measure spaces (see [14] and [11]) into X. We observe, for example, that conical singularities based on expanders (with K ≤ 1) cannot be smoothed, not even with the most generous notion of smoothing. (This will be brought into the group-theoretic framework in [8].) We furnish all necessary definitions and illustrate them by examples, but refer to textbooks for the details of standard arguments (see [2] and references therein). §1. Metrics and geodesics Given a metric space X = (X, dist), we often abbreviate and write |x − y| = |x − y| X = dist(x, y). We call X a geodesic space if every two points x and y in X can be joined, albeit nonuniquely, by a shortest (geodesic) segment [x, y] ⊂ X, i.e., an isometric embedding of a real segment of length = dist X (x, y) into X. Actually, the existence of such a shortest, or minimizing, segment is not so crucial: for most purposes, it is sufficient to have dist(x, y) equal to the infimum of the length of paths in X joining x and y, where this infimum may not be achieved. Also, we could use the middle point condition: the existence of z ∈ X such that dist(x, z) = dist(z, y) = 1 2 dist(x, y). For complete metric spaces, the latter condition is equivalent to existence of a minimizing segment. Sometimes, we could require even less: the existence of z = z ε for each ε > 0, such that both distances dist(x, z) and dist(z, y) are at most 1 2 dist(x, y) + ε. From now on, we assume the existence of our segments [x, y] ⊂ X when we deal with geodesic spaces.