The Boundary at Infinity of a CAT(0) Space
In this chapter we study the geometry at infinity of CAT(0) spaces. If X is a simply connected complete Riemannian n-manifold of non-positive curvature, then the exponential map from each point x ∈ X is a diffeomorphism onto X. At an intuitive level, one
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In this chapter we study the geometry at infinity of eAT(O) spaces. If X is a simply connected complete Riemannian n-manifold of non-positive curvature, then the exponential map from each point x E X is a diffeomorphism onto X. At an intuitive level, one might describe this by saying that, as in our own space, the field of vision of an observer at any point in X extends indefinitely through spheres of increasing radius. One obtains a natural compactification X of X by attaching to X the inverse limit of these spheres. Xis homeomorphic to a closed n-ball; the ideal points ax = X" X correspond to geodesic rays issuing from an arbitrary basepoint in X and are referred to as points at infinity. We shall generalize this construction to the case of complete eAT(O) spaces X. We shall give two constructions of ax, the first follows the visual description given above (and is due to Eberlein and O'Neill [EbON73]) and the second (described by Gromov in [BaGS85]) arises from a natural embedding of X into the space of continuous functions on X. These constructions are equivalent (Theorem 8.13). In general ax is not a sphere. If X is not locally compact then in general X and ax will not even be compact. We call X the bordification of X. Isometries of X extend uniquely to homeomorphisms of X (see (8.9». In the last section of this chapter we shall characterize parabolic isometries of complete eAT(O) spaces in terms of their fixed points at infinity.
Asymptotic Rays and the Boundary ax 8.1 Definition. Let X be a metric space. Two geodesic rays c, c' : [0,00) --+ X are said to be asymptotic if there exists a constant K such that d(c(t), c'(t» ::::: K for all t :::: O. The set ax of boundary points of X (which we shall also call the points at infinity) is the set of equivalence classes of geodesic rays - two geodesic rays being equivalent if and only if they are asymptotic. The union X u ax will be denoted X. The equivalence class of a geodesic ray c will be denoted c(oo). A typical point of ax will often be denoted ~. Notice that the images of two asymptotic geodesic rays under any isometry y of X are again asymptotic geodesic rays, and hence y extends to give a bijection of X, which we shall continue to denote by y. M. R. Bridson et al., Metric Spaces of Non-Positive Curvature © Springer-Verlag Berlin Heidelberg 1999
Asymptotic Rays and the Boundary ax
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8.2 Proposition. [fX is a complete CAT(O) space and c : [0, (0) ---+ X is a geodesic ray issuingfromx, thenforevery point x' E X there is a unique geodesic ray c' which issues from x' and is asymptotic to c. The uniqueness assertion in this proposition follows immediately from the convexity of the distance function in CAT(O) spaces (2.2). For if two asymptotic rays c' and c" issue from the same point, then d(c/(t), c"(t)) is a bounded, non-negative, convex function defined for all t 2: 0; it vanishes at 0 and hence is identically zero. In order to prove the asserted existence of c' we shall need the following lemma.
8.3 Lemma. Given I': > 0, a > 0 and s > 0, there exists a
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