This post is based on parts of Martin Bridson’s 2006 ICM article.
In the next post we will give an overview of non-positively curved groups. In preparation, here is an introduction to non-positively curved metric spaces.
CAT(0) and CAT(-1) spaces
Named by Gromov in honour of Cartan, Alexandrov and Toponogov, a CAT() condition is a notion of curvature for a geodesic space — a metric space in which every pair of points , is connected by a path of length .
The CAT() condition compares geodesic triangles in with triangles in a model space of constant curvature . It is a generalisation of a classical notion of curvature: assuming they are sufficiently smooth (such as ), then for Riemannian manifolds, locally the CAT() condition is equivalent to all the sectional curvatures being at most — see the Appendix to Chapter II.1 of Part II of Bridson and Haefliger’s book. We will limit our discussion to CAT(0) and CAT(-1) spaces; in these cases the model spaces are the Euclidean plane and the hyperbolic plane , respectively.
Here is what the CAT(0) and CAT(-1) conditions demand. Suppose is a geodesic triangle in . Let be a triangle (a comparison triangle) in (for the CAT(0) condition) or (for the stronger CAT(-1) condition) with the same side–lengths as . Suppose and are points on . Let and be the corresponding points on — that is, the sides they lie on are those that correspond to the sides and lie on, and their locations on those side agree with those of and on theirs. Then .
The usefulness of the CAT(0) and CAT(-1) conditions is greatly enhanced by the fact that they can readily be checked for suitable spaces (certain complexes) using Gromov’s link condition — more on which in some future post (probably).
We will now give two salient geometric features of a CAT(0) or CAT(-1) space .
I. Triangles in CAT(-1) spaces are uniformly thin — that is, there exists such that every geodesic triangle in is –thin, meaning that each side is in the -neighbourhood of the other two sides. This is true of CAT(-1) spaces because it holds for , where in fact can be taken to be any number greater than . As explained in Gersten’s Banff Notes, one needs only consider ideal triangles since a triangle’s vertices can be pushed out to the boundary whilst only making it less thin. Then, as ideal triangles in are equivalent, we can pick our favourite one; calculating for that triangle is a straightforward calculus problem.
II. The metric in a CAT(0) space is convex — that is, if are geodesics, parametrized proportional to arc–length, then
for all , where . Again, to see why this is true in , we look at the corresponding claim in the model space — this time, . Suppose follow opposite sides of a quadrilateral and follows the diagonal from to (all parametrized proportional to arc length). Then for all , where , since is the distance from to and is that from to . So if is a geodesic from to , then and by the CAT(0) condition, and therefore . So
Convexity has two important consequences.
- Uniqueness of geodesics. For each pairs of points in , there is a unique geodesic from one to the other.
- Contractibility. The space can be null–homotoped to a basepoint by which takes to the point on the geodesics from to a distance from . (Thus CAT(0) spaces have the potential to be universal covers of classifying spaces for discrete groups.)