The local structure of length spaces with curvature bounded above

Math. Z. 231, 409–456 (1999)

c Springer-Verlag 1999

The local structure of length spaces with curvature bounded above Bruce Kleiner? Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA (e-mail: [email protected]) Received April 21, 1998

Abstract. We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT (0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X.

1 Introduction Spaces with curvature bounded above were introduced by Alexandrov in [Ale51]; see [ABN86] for a survey of developments in the subject prior to 1980. Gromov’s paper [Gro87], led to an explosion of literature on singular spaces, see the bibliography of [Bal95]. Examples of spaces with curvature bounded above (henceforth CBA spaces) include: – Complete Riemannian manifolds with sectional curvature bounded above. – Euclidean, spherical, and hyperbolic Tits buildings, see [Tit74, Ron89, KL97]. – Complexes with piecewise constant curvature. ´ [DJ91, CD95, Ben91, BB94, S93] construct examples with interesting geometric and topological properties. ? Supported by a Sloan Foundation Fellowship, and NSF grants DMS-95-05175, DMS96-26911, DMS-9022140.

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– Limits of Hadamard spaces1 , such as Tits boundaries and asymptotic cones. These have a number of applications, see for example [Mos73, BGS85, KL95, KL97]. In this paper we will study the local geometry and topology of CBA spaces. We recall ([Nik95], [KL97, Sect. 2.1.3] or Sect. 2.1) that there is a CAT (1) space, the space of directions Σp X, associated with each point p in a CBA space X. We define the geometric dimension of CBA spaces to be the smallest function2 on the class of CBA spaces such that a) GeomDim(X) = 0 if X is discrete, and b) GeomDim(X) ≥ 1 + GeomDim(Σp X) for every p ∈ X. In other words, to find the geometric dimension of a CBA space we look for the largest number of times that we can pass to spaces of directions without getting the empty set. Our results relate this notion of dimension with several others. Theorem A Let X be a CBA space. Then the following (possibly infinite) quantities are equal to the geometric dimension of X: 1. sup{TopDim(K) | K ⊂ X is compact}. TopDim(Y ) denotes the topological dimension of Y . 2. sup{k | Hk (U, V ) 6= {0} for some open pair (U, V ) in X}. 3. sup{k | There are sequences Rj → 0, Sj ⊂ X so that d(Sj , p) → 0 for some p ∈ X, and R1j Sj converges to the unit ball B(1) ⊂ Ek in the Gromov-Hausdorff topology}. 4. sup{k | There is a biHölder embedding (with exponent 12 ) of an open set U ⊂ Ek into X}. When the geometric dimension of X is finite, we have somewhat stronger conclusions: Theorem B Let X be a CBA space, and suppose GeomDim(X) is finite. Then the following quantities are equal to GeomDim(X). 1. sup{k | Hk−1 (Σp X) 6= {0} for some p ∈ X}. 2. sup{k | Hk (X, X − p) 6= {0} for some p ∈ X}. 3. sup{k | For every > 0 there is a (1 + )-biLipschitz embedding of an open set U ⊂ Ek into X}. 4. sup{k | There is an isometric embedding of the standard sphere S k−1 (1) ⊂ Ek into Σp X, for some p ∈ X}. We remark that the Hausdorff dimension of a metric space is always at least as big as its topological dimension [HW69, Chapter VII]; however, there are Following [Bal95] we call CAT (0) spaces (complete simply connected length spaces with nonpositive curvature) Hadamard spaces. 2 Taking values in N ∪ ∞. 1

The local structure of length spaces

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compact Hadamard spaces X with GeomDim(X) = 1 which have infinite Hausdorff dimension.3 The next theorem shows that the dimension of limits of a Hadamard space X is controlled by the dimension of flats occurring in X. Theorem C Let X be a locally compact Hadamard space on which Isom(X) acts cocompactly. Then the following are equal: 1. sup{k | There is an isometric embedding Ek −→ X}. Note that this is finite. 2. sup{k | There is a quasi-isometric embedding φ : Ek −→ X}. 3. 1 + GeomDim(∂T X). Here ∂T X denotes the geometric boundary of X equipped with the Tits angle metric, see Sect. 2.4 or [KL97, Sect. 2.3.2]. 4. sup{k | Hk−1 (∂T X) 6= {0}}. 5. sup{k | There is an isometric embedding of the standard sphere S k−1 (1) ⊂ Ek in ∂T X}. 6. The geometric dimension of any asymptotic cone of X. Theorem C may be formulated for arbitrary families of Hadamard spaces, see Theorem 7.1; in particular it may be adapted to a foliated setting. Theorem C extends the main result of [AS86], see also [Gro93, pp.129-30]. Theorems A-C may also be applied to many of the questions posed in [Gro93, pp. 127-133], see Sect. 9 for a discussion. A key ingredient in the proofs of the theorems is the notion of a barycentric simplex. If X is a Hadamard space and z = (z0 , . . . , zn ) ∈ X n+1 , the barycentric simplex determined by z is the singular simplex σz : ∆n −→ X . . . , αn ) ∈ ∆n to the unique minimum of the which maps each α = (α0 ,P uniformly convex function αi [d(zi , ·)]2 . Barycentric simplices are Lipschitz, and the restriction to each face of ∆n is also a barycentric simplex (up to simplicial reparametrization). A remarkable feature of barycentric simplices which is not shared by some other constructions of simplices – such as the iterated coning of [Thu] – is that if n > GeomDim(X) then σz is degenerate: σz (∆n ) = σz (Bdy(∆n )) (see Sect. 4). Since σz is Lipschitz this implies that the image of every barycentric simplex has Hausdorff dimension ≤ GeomDim(X); this leads to the estimate TopDim(K) ≤ GeomDim(X) for compact subsets K ⊂ X. To obtain biHölder and biLipschitz embeddings of open sets U ⊂ Ek into X, we use the “nondegenerate part” of σz (∆n ), i.e. σz (∆n ) \ σz (Bdy(∆n )); this turns out to be a topological manifold in spite of the fact that σz is typically nowhere locally injective. In view of Theorem A, we make the following 3

The completion of a metric simplicial tree which “branches fast enough” will have infinite Hausdorff dimension, although it may still be compact. In this case if we remove {x ∈ X | |Σx X| = 1} we get a countably 1-rectifiable set: a set with Hausdorff dimension 1.

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Conjecture(cf. [Gro93, pp. 133]) If X is a CBA space, then TopDim(X) = GeomDim(X). When X is separable, the conjecture follows from the method of proof of Theorem A. It is interesting to compare Alexandrov spaces with curvature bounded above with Alexandrov spaces with curvature bounded below (CBB spaces). The papers [BGP92, Per94] show that a CBB space X has very restricted structure provided its Hausdorff or topological dimension is finite: [Per94] shows in particular that X is locally homeomorphic to its tangent cone at each point, and that X is a stratified manifold. This implies that the links of a polyhedron with a CBB metric are homotopy equivalent to spaces with curvature ≥ 1; and this is a very strong restriction on the polyhedron. In contrast to this, Berestovskii’s result [Ber83] shows that an upper curvature bound does not impose any restriction on topology, at least if one works in the setting of polyhedra. Also, a CBA space with finite Hausdorff dimension need not have manifold points: build an R-tree by completing an increasing union T1 ⊂ T2 ⊂ . . . of metric simplicial trees where Length(Tk ) ≤ 1 and every branch point free segment σ ⊂ Tk has length ≤ k1 ; the resulting space has Hausdorff dimension 1. There are other natural conditions that one can impose on a CBA space. If a CBA space X is locally compact and has extendible geodesics4 , then one can prove statements analogous to [BGP92, 5.4,10.6] about X; however there is a 2-dimensional locally compact Hadamard space with extendible geodesics which is not a stratified manifold, even though it is a Gromov-Hausdorff limit of a sequence of 2-dimensional CBA polyhedra which satisfy the same conditions. The natural applications of CBB and CBA spaces are also different. Finite dimensional CBB spaces arise as Gromov-Hausdorff limits of Riemannian manifolds with a lower bound on curvature and an upper bound on dimension, and are a powerful tool for studying (pre)compactness for families of Riemannian manifolds. There are two main classes of “interesting” singular CBA spaces. The first one is locally finite polyhedra with nonpositively curved metrics; these provide numerous examples with interesting topology and fundamental groups. The second is Tits boundaries and asymptotic cones – these are limit spaces associated with a Hadamard space, and are usually “large”, i.e. nonseparable. The Tits boundary of a Hadamard space X is an important tool for studying the isometry group and the geodesic flow of X. Asymptotic cones arise in compactness arguments, for instance when studying degeneration of hyperbolic structures [MS84,Pau88,RS94, Bes88], or when studying quasi-isometries [KL95,KL97,KKL98]. We recall that a complete length space (X, d) is convex [Gro78] if for every pair γ1 : [a1 , b1 ] → X and γ2 : [a2 , b2 ] → X of constant speed 4

These assumptions are natural when studying the geodesic flow.


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geodesic segments, the function d ◦ (γ1 , γ2 ) : [a1 , b1 ] × [a2 , b2 ] → R is convex. It turns out that a slightly weaker version of Theorem C holds for the more general class of convex length spaces. Theorem D Let X be a locally compact convex length space with cocompact isometry group. Then the following are equal: 1. sup{k | There is an isometric embedding of a k-dimensional Banach space in X}. Note that this is finite. 2. sup{k | There is a quasi-isometric embedding φ : Ek → X}. 3. sup{k | Hk (U, V ) 6= {0} for some open pair V ⊆ U ⊆ CT X}. CT X denotes the Tits cone of X, see Sect. 10 for the definition. 4. sup{k | There is a k-dimensional Banach space (Rk , k · k), sequences Rj → 0, Sj ⊂ CT X so that R1j Sj converges to the unit ball in (Rk , k· k) in the Gromov-Hausdorff topology}. 5. sup{k | There is an asymptotic cone Xω of X, and an open pair V ⊆ U ⊆ Xω such that Hk (U, V ) 6= {0}}. 6. sup{k | There is a k-dimensional Banach space (Rk , k · k), sequences Rj → 0, Sj ⊂ Xω so that R1j Sj converges to the unit ball in (Rk , k · k) in the Gromov-Hausdorff topology}. 7. sup{k | There is a k-dimensional Banach space (Rk , k · k), sequences Rj → ∞, Sj ⊂ X so that R1j Sj converges to the unit ball in (Rk , k · k) in the Gromov-Hausdorff topology}. The proof of Theorem D is somewhat different from the proof of Theorem C because barycentric simplices do not behave well in convex spaces (squared distance functions are no longer uniformly convex). Instead we use the differentiation theory for Lipschitz maps into metric spaces of KorevaarSchoen [KS93] (see Theorem 10.7, Corollary 10.9, and Proposition 10.18). The paper is organized as follows. In Sect. 2 we briefly recall background material and introduce notation. In Sect. 3 we use the geometric dimension to bound topological and homological dimensions from below. In Sect. 4 we define barycentric simplices, and prove that they degenerate above the geometric dimension (Corollary 4.10). In Sect. 5 we use the results of Sect. 4 to prove that the geometric dimension is at least as big as the topological and homological dimensions. In Sect. 6 we construct biHölder and biLipschitz embeddings of open sets U ⊆ En into CBA spaces. In Sect. 7 we construct flats in Hadamard spaces X starting with objects in ∂T X or in an asymptotic cone of X. In Sect. 8 we combine the earlier sections to prove Theorems A, B, and C. In Sect. 9 we discuss questions posed in [Gro93]. In Sect. 10 we discuss convex length spaces, giving a short proof of (a special case of) the metric differentiation result of [KS93] and applications to convex length spaces.

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I would like to thank Werner Ballmann, Toby Colding, Chris Croke, Christoph Hummel, and Bernhard Leeb for their interest in this work. I would especially like to thank Werner Ballmann and Christoph Hummel for making numerous helpful remarks on an earlier version of this paper. Theorems A-C (except for part 4 of Theorem B) were obtained when I was visiting MSRI in 1993-94, using somewhat different arguments. Contents 1 2 3 4 5 6 7 8 9 10

Introduction . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . Geometric dimension bounds topological and homological dimension . . . . . . . . . . . . . . . Barycentric simplices . . . . . . . . . . . . . . . . . . Topological dimension . . . . . . . . . . . . . . . . . . Minimum sets and subsets homeomorphic to Rn . . . . The Tits boundary, asymptotic cones, flats, and the geometric dimension . . . . . . . . . . . . . . Proofs of Theorems A, B, and C . . . . . . . . . . . . . Questions from Asymptotic Invariants of infinite groups Convex length spaces . . . . . . . . . . . . . . . . . .

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2 Preliminaries We refer the reader to [KL97] for a more detailed discussion of the material in this section. 2.1 CAT (κ) spaces, spaces of directions, tangent cones, etc Let Mκ2 denote the two-dimensional model space with constant curvature κ, and let D(κ) denote the diameter of Mκ2 . A complete metric space X is a CAT (κ) space if 1. Diam(X) ≤ D(κ). 2. Every two points x1 , x2 ∈ X with d(x1 , x2 ) < D(κ) are joined by a geodesic segment of length d(x1 , x2 ). Note that some other definitions in the literature require X to be a length space; this is inconvenient in induction arguments since spaces of directions can be disconnected. 3. Every geodesic triangle in X with perimeter < 2D(κ) is at least as thin as a geodesic triangle in Mκ2 with the corresponding side lengths. In particular, every two points x1 , x2 ∈ X with d(x1 , x2 ) < D(κ) are joined by a unique geodesic segment with length < D(κ) (up to reparametrization), which we will confuse with its image x1 x2 .


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Let X be a CAT (κ) space. If p, x, y ∈ X and d(p, x) + d(x, y) + d(p, y) < 2D(κ), then there is a well-defined geodesic triangle ∆pxy. The comparison angle of the triangle ∆pxy at p is defined to be the angle of the comparison triangle (in Mκ2 ) for ∆pxy at the vertex corresponding to e p (x, y). The thin triangle condition implies that if p; this angle is denoted ∠ 0 0 e p (x0 , y 0 ) ≤ ∠ e p (x, y). Therefore x ∈ px − {p} and y ∈ py − {p} then ∠ e p (x0 , y 0 ) has a limit; we call if we let x0 ∈ px, y 0 ∈ py tend to p then ∠ this the angle between px and py at p, and denote it by ∠p (x, y). If we let e p (x, y 0 ) also tends to ∠p (x, y). The function y 0 ∈ py − {p} tend to p, then ∠ p 7→ ∠p (x, y) is upper semicontinuous. ∠p defines a pseudo-metric (see definition 2.7) on the collection of geodesic segments leaving p. We define Σp∗ X to be the associated quotient metric space obtained by collapsing zero diameter subsets to points. The space of directions at p is the completion of Σp∗ X, and is denoted Σp X. Σp X is a CAT (1) space [Nik95] (see also [KL97, Sect. 2.1.3]). The tangent cone at p is the Euclidean cone C(Σp X) over Σp X; it is a CAT (0)-space and is denoted by Cp X. We will often identify Σp X (as a set) with the points in Cp X at unit distance from the vertex of the cone. We have a map logΣp X : B(p, D(κ)) − {p} −→ Σp X which takes x ∈ B(p, D(κ)) − {p} to the direction of the segment px. We → for log will often use the notation − px Σp X x. There is also a map logCp X : B(p, D(κ)) −→ Cp X which takes x ∈ B(p, D(κ)) to the unique point on the ray C(logΣp X (x)) at distance d(p, x) from the vertex. It follows from comparison inequalities that logCp X is 1-Lipschitz when κ ≤ 0; more generally, if r < D(κ) then logCp X |B(p,r) is L(r, κ)-Lipschitz.

Lemma 2.1 If X is a CAT (κ) space, p ∈ X, and K ⊂ Cp X is a compact subset, then there are sequences Rj ∈ R, Sj ⊂ X so that limj→∞ Rj = 0, limj→∞ d(Sj , p) = 0, and R1j Sj converges to K in the Gromov-Hausdorff topology. Proof. First assume K = logCp X (Y ) where Y ⊂ B(p, D(κ)) is a finite set. For each y ∈ Y let γy : [0, 1] → X be the constant speed geodesic with γy (0) = p, γy (1) = y. Then d(γy1 (t), γy2 (t)) = d logCp X (y1 ), logCp X (y2 ) , lim t→0 t so if we set Sj := {γy ( 1j ) | y ∈ Y } then R1j Sj → K in the GromovHausdorff topology. Since Σp∗ X is dense in Σp X, the Euclidean cone C(Σp∗ X) is dense in Cp X := C(Σp X); therefore if K ⊂ Cp X is an arbitrary compact set then K is a Hausdorff limit of a sequence Ki ⊂ C(Σp∗ X) ⊆ Cp X of finite subsets. For each i there is a λi ∈ (0, ∞) so that λi Ki (the image of Ki

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under the λi homothety of the cone Cp X) is in Im(logCp X ). Applying the preceding paragraph to each Ki and a diagonal construction we obtain the desired sequences. t u A metric space X has curvature bounded above by κ if for every p ∈ X there is an r > 0 so that the closed ball Bp (r) is a CAT (κ) space; a space X has curvature bounded above (or is a CBA space) if it has curvature bounded above by κ for some κ. If X is a CBA space, p ∈ X and r > 0 is small enough that Bp (r) ⊆ X is a CAT (κ) space, then we may apply all the constructions of the previous paragraph to Bp (r). One observes that Σp X and Cp X are independent of the choice of r, κ. We define the injectivity radius at p, Inj(p), to be the supremum of the radii r so that Bp (r) is a CAT (κ) space for some κ with D(κ) ≥ r. We have well-defined logarithm maps logΣp X : Bp (Inj(p)) − {p} → Σp X and logCp X : Bp (Inj(p)) → Cp X. 2.2 Convex functions Definition 2.2 Let X be a complete metric space. If C ∈ R, a function f : X −→ R is C-convex if for every unit speed geodesic γ : [a, b] −→ X, the function t 7→ f ◦ γ(t) − C2 t2 is convex. In particular, if f is C-convex, then f ◦ γ has left and right derivatives for every geodesic γ. Lemma 2.3 Let X be a complete metric space such that every two points in X are joined by a geodesic segment. Suppose C > 0, and f : X → R is a continuous C-convex function which is bounded below. Then f has a unique minimum x ¯ and any sequence xk ∈ X with limk→∞ (f (xk )) = inf f converges to x ¯. Proof. Suppose xk is a sequence with f (xk ) → inf f as k → ∞. For every k, l let zkl be the midpoint of a geodesic segment joining xk to xl . Then C-convexity of f gives 1 1 f (zkl ) ≤ f (xk ) + f (xl ) − [d(xk , xl )]2 . 2 2 8 Hence limk,l→∞ d(xk , xl ) = 0 and xk is Cauchy. Then limk→∞ xk is the unique minimum of f . u t Lemma 2.4 (Directional derivatives of convex functions) Let X be a space with curvature bounded above, and let φ : X −→ R be an L-Lipschitz function on X which is C-convex for some C ∈ R. For every p ∈ X there is a unique L-Lipschitz function Dφ : Cp X −→ R such that for every x ∈ X − {p} in the domain of logCp X , Dφ(logCp X (x)) = (φ ◦ γ)0 (0)


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where γ : [0, 1] → X is the constant speed parametrization of the segment px. Moreover Dφ is convex and homogeneous of degree 1: (Dφ)(λv) = λ(Dφ)(v) for every λ ∈ [0, ∞). Proof. Replacing X with a closed convex ball centered at p, we may assume that logCp X is defined on all of X. Define ψ : X −→ R by setting ψ(x) = (φ ◦ γx )0 (0) where γx : [0, 1] → X is the constant speed geodesic segment with γx (0) = p, and γx (1) = x. If x1 , x2 ∈ X, then |ψ(x1 ) − ψ(x2 )| = |(φ ◦ γx1 )0 (0) − (φ ◦ γx2 )0 (0)| (φ ◦ γx1 )(t) − (φ ◦ γx2 )(t) = lim t→0 t d(γx1 (t), γx2 (t)) ≤ L lim t→0 t = Ld(logCp X (x1 ), logCp X (x2 )). Therefore ψ descends to an L-Lipschitz function Dφ : Cp X → R which is homogeneous of degree 1. The convexity of Dφ follows from the Cconvexity of φ. t u A computation of Hessians shows that if p is in the model space Mκ2 , then the distance function dp := d(p, ·) is C1 (r, κ)-convex on Bp (r) when r < D(κ). By triangle comparison this implies that if X is a CAT (κ) space and p ∈ X, then dp is a C1 (r, κ)-convex function on Bp (r) when r < D(κ). The directional derivative of dp at x ∈ Bp (D(κ)) − {p} is given by → = −hv, log xpi (Ddp )(v) = −hv, − Σx X pi

(2.5)

where the “inner product” is defined by hv, wix := |v||w| cos ∠x (v, w). Similarly, there is a function C2 (r, κ) > 0 so that the squared distance function d2p is C2 (r, κ)-convex on Bp (r) when r < D(κ) 2 . 2.3 Ultralimits and asymptotic cones A nonprincipal ultrafilter on N is a finitely additive probability measure ω on N so that ω(S) ∈ {0, 1} for every subset S ⊆ N, and ω(S) = 0 when S ⊂ N is finite. If K is a compact metric space and f : N → K, then there is a unique point p ∈ K with the property that for every neighborhood U of p, ω(f −1 (U )) = 1; this point is called the ω-limit of f and is denoted f (ω)

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or ω-lim f . If {(Xi , di , ?i )}i∈N is a sequence of pointed metric spaces, we define a pseudo-metric (see definition 2.7) dˆω on ( (xi ) ∈

Y i

) Xi | sup di (xi , ?i ) < ∞ i

⊆

Y

Xi

i

by dˆω ((xi ), (yi )) := ω-lim di (xi , yi ). We let Xω denote the associated quotient metric space with distance function dω , and let ?ω ∈ Xω denote the image of (?i ) under the projection. The ultralimit of (Xi , di , ?i ) is the pointed metric space (Xω , dω , ?ω ); we will also use the notation ω-lim(Xi , di , ?i ) and sometimes suppress di when it is clear from the context. Properties of ultralimits: 1. Ultralimits are complete metric spaces. 2. If (Xi , di , ?i ) is a Gromov-Hausdorff precompact sequence of spaces, then (Xω , dω , ?ω ) is a limit point of the sequence. 3. If {(Xi , di , ?i )}i∈N is a sequence of pointed metric spaces and every closed ball in (Xω , dω ) is compact, then there is a subset S ⊆ N with ω(S) = 1 so that the corresponding subsequence of {(Xi , di , ?i )}i∈N converges in the Gromov-Hausdorff topology to ω-lim(Xi , di , ?i ). ¯ i , d¯i , ¯?i ) is a sequence of (Li , Ai ) quasi4. If φi : (Xi , di , ?i ) → (X isometric embeddings (resp. quasi-isometries), ω-lim di (φi (¯?i ), ?i ) < ∞, ω-lim Li = L < ∞, ω-lim Ai = A < ∞, then we get induced quasi¯ ω , d¯ω ); isometric embeddings (resp. quasi-isometries) φω : (Xω , dω ) → (X when A = 0 then φω is uniquely determined. 5. If K ⊂ (Xω , dω , ?ω ) is a compact subset, then there is a sequence (Ki , d¯i , ¯?i ) of pointed finite metric spaces and a sequence of isometric embeddings φi : (Ki , d¯i , ¯?i ) → (Xi , di , ?i ) so that φω : Kω → Xω maps Kω isometrically onto K. 6. An ultralimit of a sequence of CAT (κ) spaces is a CAT (κ) space. Let (X, d) be a metric space. An asymptotic cone of X is an ultralimit of the form ω-lim(X, λi d, ?i ) (sometimes written (λi X, ?i ) when the metric is clear from the context) where ?i ∈ X, λi > 0, and ω-lim λi = 0. Properties of asymptotic cones: ¯ is an (L, A) quasi-isometric embedding (resp. quasi1. If φ : X → X isometry), then the ultralimit of the sequence ¯ φ(?i )) ¯ λi d, φ : (X, λi d, ?i ) → (X, gives an L-biLipschitz embedding (resp. L-biLipschitz homeomorphism) between the asymptotic cones. 2. If the isometry group of X acts with cobounded orbits on X, then the isometry group of any asymptotic cone of X acts transitively.


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2.4 The Geometric and Tits boundaries Let X be a Hadamard space. Two unit speed geodesics5 γ1 : [0, ∞) → X, γ2 : [0, ∞) → X are equivalent (or asymptotic) if the Hausdorff distance between γ1 ([0, ∞)) and γ2 ([0, ∞)) is finite. Given a geodesic ray γ1 and p ∈ X, there is a unique geodesic ray γ2 : [0, ∞) → X asymptotic to γ1 with γ2 (0) = p. The geometric boundary of X, ∂∞ X, is the set of equivalence classes of geodesic rays in X topologized by viewing it as the collection of geodesic rays leaving some p ∈ X endowed with the compactopen topology. If p ∈ X, ξ ∈ ∂∞ X, we let pξ denote the image of the geodesic ray leaving p in the class ξ. The Tits angle between two geodesic ray γ1 , γ2 leaving p ∈ X is e p (γ1 (t), γ2 (t)). ∠T defines a metric on ∂∞ X, ∠T (γ1 , γ2 ) := limt→∞ ∠ and we denote the resulting metric space by ∂T X. Properties of ∂T X: 1. ∂T X is CAT (1)-space. 2. ∠T : ∂∞ X × ∂∞ X → [0, π] is a lower-semicontinuous function. 3. If p ∈ X, ξ1 , ξ2 ∈ ∂T X, then ∠T (ξ1 , ξ2 ) = supx∈X ∠x (ξ1 , ξ2 ) = limx∈pξ1 ,d(x,p)→∞ ∠x (ξ1 , ξ2 ). 2.5 Miscellany Definition 2.6 An -Hausdorff approximation is a map φ between metric spaces (X, dX ) and (Y, dY ) so that |φ∗ dY − dX | < and dY (y, φ(X)) < for every y ∈ Y . A sequence of metric spaces Xk converges to a metric space X in the Gromov-Hausdorff topology iff there is a sequence of k -Hausdorff approximations φk : X → Xk with limk→∞ k = 0. Definition 2.7 A pseudo-metric, or pseudo-distance function on a set X is a function d : X × X → [0, ∞) which is symmetric and which satisfies the triangle inequality. ¯ by let¯ d) If (X, d) is a pseudo-metric space, then we get a metric space (X, ¯ ¯ ting X be the set of maximal zero diameter subsets and setting d(S1 , S2 ) := d(s1 , s2 ) for any si ∈ Si . We define the Gromov-Hausdorff pseudo-distance between pseudo-metric spaces to be the Gromov-Hausdorff distance between the associated (quotient) metric spaces. 3 Geometric dimension bounds topological and homological dimension In this section we show that the geometric dimension gives a lower bound for the topological and homological dimensions of a CBA space. 5

Geodesic rays will be parametrized at unit speed except in Sect. 10.

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Lemma 3.1 If X is a CBA space, p ∈ X, and GeomDim(Σp X) ≥ n − 1, then there are sequences Rj → 0, Sj ⊂ X so that d(Sj , p) → 0 and R1j Sj converges to the unit ball B n ⊂ En in the Gromov-Hausdorff topology. Proof. The statement is immediate when n = 0, so we assume that n > 0. By induction there is a sequence of sets Tk ⊂ Σp X so that R¯1 Tk → B n−1 ⊂ k En−1 in the Gromov-Hausdorff topology. If CTk ⊆ Cp X is the Euclidean cone over Tk , and ?k ∈ CTk ∩ Σp X, then the pointed spaces ( R¯1 CTk , ?k ) k converge to B n−1 × R in the pointed Gromov-Hausdorff topology. Since B n embeds isometrically in B n−1 × R we may find T¯k ⊂ CTk so that ( R1k T¯k , ?k ) converges to B n . But by Lemma 2.1 each T¯k is a GromovHausdorff limit of a sequence R10 Sj0 where Sj0 ⊂ X, d(Sj0 , p) → 0. Hence j

the lemma.

t u

Proposition 3.2 If X is a CBA space, p ∈ X, Rk → 0, Sk ⊆ X, d(Sk , p) → 0, and R1k Sk → B n ⊂ En in the Gromov-Hausdorff topology, then Hn (X, V ) 6= 0 for some open set V ⊂ X and TopDim(K) ≥ n for some compact set K ⊆ X. Proof. Let ei be the ith unit coordinate vector in En , and let α : En −→ En be the map with ith coordinate function d(ei , ·) − 1. Note that if δ ∈ (0, 1) is sufficiently small, then α(∂B(0, δ)) ⊂ En \ {0} and α|B(0,δ) induces an isomorphism Hn (B(0, δ), ∂B(0, δ)) −→ Hn (En , En \ {0}). Let φk : B(0, Rk ) −→ Sk ⊆ X be a sequence of k Rk -Hausdorff approximations (see definition 2.6), where k → 0. Set xik = φk (Rk ei ), and define dk : X −→ En by dk (·) = ( R1k (d(x1k , ·) − Rk ), . . . , R1k (d(xnk , ·) − Rk )). Triangulate B(0, Rk ) ⊂ En with smooth simplices of diameter < k Rk . For sufficiently large k define a continuous map ψk : B(0, Rk ) −→ X by starting with the restriction of φk to the 0-skeleton of the triangulation, and extending it to B(0, Rk ) using barycentric simplices (see Sect. 4). Define αk : B(0, δ) −→ En by αk (x) = (dk ◦ ψk )(Rk x). Then αk converges uniformly to α|B(0,δ) and for sufficiently large k we have αk (∂B(0, δ)) ⊂ En − \{0} and Hn (αk ) : Hn (B(0, δ), ∂B(0, δ)) → Hn (En , En \ {0}) is n an isomorphism. Hence if Vk = d−1 k (E \ {0}), we find that for k sufficiently large Hn (ψk ) : Hn (B(0, δRk ), ∂B(0, δRk )) −→ Hn (X, Vk ) is well-defined and nontrivial. So for sufficiently large k, dk |ψ (B(0,δR )) cank k not be uniformly approximated by continuous maps which miss the origin 0 ∈ En , and hence TopDim(ψk (B(0, δRk ))) ≥ n. t u Remark. The proof shows that there are , R > 0 depending on κ ∈ R and n ∈ N so that if B n ⊂ En denotes the unit ball, and a CAT (κ)-space X contains a subset S with dGH ( R10 S, B n ) < , R0 < R, then TopDim(X) ≥


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n. When κ ≤ 0 one may take R = ∞. In fact, it suffices to have a (sufficiently small) set which is sufficiently close in the Gromov-Hausdorff topology to either a) the 0-skeleton of the first barycentric subdivision of a regular nsimplex or b) the set {0, ±R0 e1 , . . . , ±R0 en } ⊂ En . 4 Barycentric simplices In this section we define a notion of simplex which provides a direct connection between the geometric dimension of a CBA space and the topological dimension of its separable subsets. The key result is Proposition 4.8, which implies that if GeomDim(X) ≤ n, then every barycentric k-simplex with k > n is degenerate (see definition 4.7). Let X be a space with curvature bounded above. Definition 4.1 We say that z = (z0 , . . . , zn ) ∈ X n+1 is (κ, r)-admissible if κ ∈ R, r < D(κ) 2 , the closed ball B(zi , r) is a CAT (κ) space, and d(zi , zj ) ≤ r for 0 ≤ i, j ≤ n. z is admissible if it is (κ, r)-admissible for some (κ, r). Lemma 4.2 Let z = (z0 , . . . , zn ) ∈ X n+1 be (κ, r)-admissible, and let Yr be the CAT (κ) space ∩i B(zi , r). Then for every P α in the standard nsimplexP ∆n = {(α0 , . . . , αn ) ∈ Rn+1 | αi ≥ 0, αi = 1}, the function αi d2zi has a unique minimumPpoint in Yr . This minimum point y φα := is characterized by the property that αi dzi (y)D(dzi ) is a nonnegative function on Cy X, where D(dzi ) : Cy X −→ R denotes the directional derivative of dzi , see Lemma 2.4. Denoting this minimum point by σz (α), we obtain a Lipschitz map σz : ∆n −→ X. The map σz is independent of the choice of (κ, r), and we call it the barycentric simplex determined by z. If τ : ∆k −→ ∆n is a k-face of ∆n , then σz ◦ τ coincides with the barycentric simplex determined by the sub-(k + 1)-tuple of z selected by τ . Proof. Existence of a minimum of φα . Since z is (κ, r)-admissible, the squared distance functions d2zi are all C-convex on Yr , where C > 0 depends on κ, r (see Sect. 2). Therefore φα is C-convex on Yr for every α ∈ ∆n , and so we may apply Lemma 2.3 to see that φα has a unique minimum in Yr , and that this minimum does not depend on the choice of (κ, r). σz is Lipschitz. Pick y ∈ Yr , and let γ : [0, l] −→ Yr be the unit speed geodesic from σz (α) to y. The function φα ◦γ : [0, l] −→ R is C-convex with a minimum at 0, so the left derivate of φα ◦ γ at l is at least P Cl; in particular, if y 6= σz (α) then the derivative of φα at y, D(φα ) = 2αi dzi (y)D(dzi ), attains negative values. On the other hand, if y = σz (α0 ) for some α0 ∈ ∆n , then the left derivate of φα0 ◦ γ at l is ≤ 0 since σz (α0 ) minimizes φα0 . Therefore the left derivate of (φα − φα0 ) ◦ γ at l is ≥ Cl. d2zi is 2rLipschitz on Yr , so φα − φα0 is 2rkα − α0 kl1 -Lipschitz on Yr . This gives

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0 Cl ≤ 2rkα − α0 kl1 or d(σz (α), σz (α0 )) = l ≤ 2r C kα − α kl1 , and so σz is Lipschitz. The final statement of the lemma is immediate from the definition of σz . t u

We will need to define maps by using the minima of more general families of functions. The following lemma provides an appropriate setup. Lemma 4.3 Let X be a complete metric space. Let f0 , . . . , fn be bounded n+1 , and for every α ∈ Rn+1 define Lipschitz functions on X, let PΩ ⊂ R αi fi . Suppose ν : Ω −→ R is a continuous φα : X −→ R by φα := positive function such that for every α ∈ Ω 6 1. φ−1 α ((−∞, ν(α) + inf φα )) ⊂ X is a geodesically convex subset . −1 2. φα is ν(α)-convex on φα ((−∞, ν(α) + inf φα )). Then 1. For each α ∈ Ω, φα attains a minimum at a unique point in X. 2. If σf : Ω −→ X is the map taking each α ∈ Ω to the minimum of φα , then σf is locally Lipschitz. If in addition a) X has curvature bounded above, b) the fi ’s are all Cconvex for some C ∈ R, c) Dfi : Cx X −→ R denotes the derivative of fi at x (see Lemma 2.4), and d) α ∈ Ω; then P 3. x ∈ X is the minimum of φα iff αi Dfi is a nonnegative function on Cx X. We omit the proof as it is similar to the proof of Lemma 4.2. Lemma 4.4 Let X be a CAT (1) space, let CX be the Euclidean cone over X, and identify X with the unit sphere in CX. Let f : X −→ R be a Lipschitz function whose homogeneous extension fˆ : CX −→ R is convex. Then 1. If γ : [0, θ] −→ X is a unit speed geodesic, then (f ◦ γ)00 + f ◦ γ ≥ 0 in the sense of distributions. Equivalently, for every t ∈ (0, θ) and every > 0 there is a δ > 0 so that f ◦ γ is (−(f ◦ γ)(t) − )-convex in the interval [t − δ, t + δ] ⊂ [0, θ]. 2. For every µ > 0, f −1 ((−∞, −µ)) is a geodesically convex subset of X, and f is µ-convex on f −1 ((−∞, −µ)). Proof. 1. We may take X = γ([0, θ]) and θ ∈ (0, π). Then we may identify CX with a sector in R2 . The first statement is obvious for smooth functions and follows for general functions by a smoothing argument. 2. If x1 , x2 ∈ X and f (x1 ), f (x2 ) < 0 then the segment joining x1 to x2 in the Hadamard space CX cannot pass through the vertex o of CX, since convexity of fˆ would force fˆ(0) < 0, which is absurd. Therefore 6 Every two points are joined by path in the subset whose length equals the distance between them.


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dX (x1 , x2 ) < π, and f −1 ((−∞, 0)) is geodesically convex. The second statement now follows from the first. t u Lemma 4.5 If f0 , . . . , fn are functions on a CAT (1) space X which satisfy the conditions of Lemma 4.4, then together they satisfy the conditions of Lemma 4.3 for Ω := {α ∈ Rn+1 | αi ≥ 0, inf φα < 0} and a suitable function ν. Proof. First note that the |fi |’s are bounded by some B since X has diameter ≤ π and fi is Lipschitz; by 1 of Lemma 4.4 this tells us that the P fi are (−B)convex. For α ∈ Ω, the nonnegative linear combination αi fi satisfies ((−∞, −µ)) is the hypotheses of Lemma 4.4, so for every µ > 0, φ−1 α geodesically convex, and φα is µ-convex on φ−1 ((−∞, −µ)). So if we set α 1 ν(α) = − 2 inf φα then conditions 1 and 2 of Lemma 4.3 will be satisfied. t u Definition 4.6 If f0 , . . . , fn are functions on a CBA space X, then (f0 , . . . , fn ) is an admissible (n + 1)-tuple if 1. Each fi is C-convex for some C ∈ R. 2. f0 , . . . , fn satisfy the conditions of Lemma 4.3 with with Ω = ∆n and a suitable function ν. We call the map σf : ∆n −→ X constructed in Lemma 4.3 the simplex determined by (f0 , . . . , fn ). Note that a (κ, r)-admissible (n+1)-tuple z = (z0 , . . . , zn ) ∈ X n+1 defines an admissible (n + 1)-tuple (d2z0 , . . . , d2zn ) on Yr := ∩i B(zi , r) whose simplex is just the barycentric simplex of z. Definition 4.7 Let (f0 , . . . , fn ) be admissible. Then σf : ∆n −→ X is degenerate if σf (∆n ) = σf (Bdy(∆n )). Otherwise σf is nondegenerate, and any x ∈ σf (∆n ) − σf (Bdy(∆n )) is a nondegenerate point. When z ∈ X n+1 is admissible then we will apply the same terminology to the barycentric simplex σz : ∆n → X. Note that by part 3 of Lemma 4.3 it follows that x ∈ X is a nondegenerate point of σf iff 1. There is an α ¯ ∈ ∆n so that Dφα : Σx X → R is nonnegative. 2. For every α ∈ Bdy(∆n ) we have inf Σx X Dφα < 0. Proposition 4.8 Let X be a CBA space. If (f0 , . . . , fn ) is admissible and x ∈ σf (∆n ) is a nondegenerate point then GeomDim(Σx X) ≥ n − 1. In particular if z = (z0 , . . . , zn ) ∈ X n+1 is admissible and x ∈ σz (∆n ) is nondegenerate then GeomDim(Σx X) ≥ n − 1. Proof. The idea of the proof is to construct a Lipschitz map σg : Bdy(∆n ) → Σx X which indicates, for eachPα ∈ Bdy(∆n ), the direction of fastest deαi fi . We then construct a Lipschitz map crease for the function φα = g : Σx X → Rn+1 with the property that g ◦ σg maps the fundamental cycle of Bdy(∆n ) to a cycle in Rn+1 which is nontrivial in Hn−1 (W ) where

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W ⊂ Rn+1 is suitably chosen. It then follows that g ◦ σg (Bdy(∆n )) has positive (n − 1)-dimensional Hausdorff measure, which means that at least one of the top-dimensional faces of σg is nondegenerate. We then argue by induction, concluding that GeomDim(Σx X) ≥ n − 1. ¯ n ) ∈ ∆n so that x = σf (¯ α). Let gi := Dfi : Pick α ¯ = (¯ α0 , . . . , α let g := (g0 , . . . , gn ) : Σx X −→ R be the directional derivative of fi at x, andP Σx X −→ Rn+1 . By 3 of Lemma 4.3 we know that α ¯ i gi is nonnegative on Σx X; in particular, for every v ∈ Σx X we have P gi (v) ≥ 0 for some i. As x is a nondegenerate point of σf , we have inf αi gi < 0 for every α ∈ Bdy(∆n ). In view of Lemma 4.5, we may apply Lemma 4.3 to g0 , . . . , gn with Ω = Bdy(∆n ), to get a Lipschitz map σg : Bdy(∆n ) −→ Σx X. The composition ρ = g ◦σg : Bdy(∆n ) −→ Rn+1 has the following properties: – For every α ∈ Bdy(∆n ), ρ(α) has at least one nonnegative coordinate and at least one negative coordinate. – If I ⊂ {0, . . . , n}, and α ∈ Bdy(∆n ) lies in the face corresponding to I (i.e. αj = 0 for j 6∈ I), then the ith coordinate of ρ(α) is negative for some i ∈ I. Set W = {t ∈ Rn+1 | There exists i so that ti < 0 and j so that tj ≥ 0}, and for each subset I ⊂ {0, . . . , n}, set WI = {t ∈ W | ti < 0 for some i ∈ I}. Lemma 4.9 1. The map −id|Bdy(∆n ) : Bdy(∆n ) −→ W is a homotopy equivalence. 2. WI is contractible when I is a proper subset of {0, . . . , n}. 3. If u : Bdy(∆n ) → W is a continuous map with Hn−1 (u) : Hn−1 (Bdy(∆n )) → Hn−1 (W ) nonzero then u(Bdy(∆n )) ⊂ W has positive (n − 1)-dimensional Hausdorff measure. Proof. Let Rn+1 := {t ∈ Rn+1 | ti ≥ 0 for all i}, Rn+1 := {t ∈ Rn+1 | > ≥ ti > 0 for all i} be the nonpositive and positive orthants respectively; and and Rn+1 similarly. We have W = Rn+1 \ (Rn+1 ∪ Rn+1 define Rn+1 < < ). ≤ ≥ Note that WI deformation retracts to its intersection with Bdy(Rn+1 ≤ ) = n+1 n+1 R≤ \ R< by moving points in the (−1, . . . , −1) direction until they n+1 hit Bdy(Rn+1 ≤ ); moreover this retraction rI : WI → WI ∩ Bdy(R≤ ) is Lipschitz. Also, WI ∩ Bdy(Rn+1 ≤ ) deformation retracts to its intersection with −Bdy(∆n ) via the (locally Lipschitz) radial retraction rI0 : WI ∩ Bdy(Rn+1 ≤ ) → WI ∩ (−Bdy(∆n )). But WI ∩ (−Bdy(∆n )) is the image of the I-face of Bdy(∆n ) under −idRn+1 when I is a proper subset of {0, . . . , n} and −Bdy(∆n ) otherwise, so 1 and 2 follow. 0 If u : Bdy(∆n ) → W is continuous and Hn−1 (u) 6= 0 then r{0,... ,n} ◦ r{0,... ,n} ◦ u : Bdy(∆n ) → −Bdy(∆n ) is nontrivial in Hn−1 , so it is


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0 surjective; since r{0,... ,n} and r{0,... ,n} are both locally Lipschitz this implies that the (n−1)-dimensional Hausdorff measure of u(Bdy(∆n )) is positive. This completes the proof of Lemma 4.9. t u

We have shown that the Lipschitz map ρ : Bdy(∆n ) −→ Rn+1 has image contained in W , and the image of the I-face of Bdy(∆n ) under ρ is contained in the contractible subset WI . Since ρ : Bdy(∆n ) −→ W is homotopic to any other continuous map satisfying these conditions, ρ is homotopic to −idRn+1 |Bdy∆n ; in particular ρ maps the fundamental class of Bdy(∆n ) to a nontrivial element of Hn−1 (W ). By 3 of Lemma 4.9 we conclude that ρ(Bdy(∆n )) – and hence also σg (Bdy(∆n )) – has positive (n − 1)-dimensional Hausdorff measure. This means that for some i ∈ {0, . . . , n}, the image of the (n − 1)-face {t ∈ Bdy(∆n ) | ti = 0} of Bdy(∆n ) under σg has positive (n − 1)-dimensional Hausdorff measure. Then (g0 , . . . , gi−1 , gi+1 , . . . , gn ) is an admissible n-tuple whose simplex ∆n−1 → Σx X is nondegenerate, for its image has positive (n − 1)dimensional Hausdorff measure. By induction we conclude that GeomDim (Σx X) ≥ n − 1. t u Corollary 4.10 If X is a CBA space with GeomDim(X) = n < ∞, then for every admissible (k + 1)-tuple (f0 , . . . , fk ), the image of σf has finite n-dimensional Hausdorff measure. Proof. By Proposition 4.8, each face of σf of dimension > n is degenerate, so σf (∆k ) is the image of the n-skeleton of ∆k . As σf is Lipschitz, Corollary 4.10 follows. t u Lemma 4.11 If X is a CBA space with GeomDim(X) ≥ n, then there is an admissible z ∈ X n+1 with σz nondegenerate. Proof. Pick p ∈ X so that GeomDim(Σp X) ≥ n. By Lemma 3.1 we may find sequences Rk → 0 and Sk ⊂ X satisfying the hypotheses of Proposition 3.2. Following the proof of Proposition 3.2, the map Ψk : B(0, Rk ) → X is constructed using barycentric simplices and TopDim(Ψk (B(0, Rk ))) ≥ n. But this forces the image of one of the barycentric simplices σ to have topological dimension at least n; therefore [HW69, Chapter VII] Image(σ) has Hausdorff dimension ≥ n, and so σ is nondegenerate. Alternatively one can show directly that if the (n + 1)-tuple z k := k (z0 , . . . , znk ) ∈ [B(0, Rk )]n+1 consists of the vertices of a regular n-simplex inscribed in B(0, Rk ), then φk (z k ) = (φk (z0k ), . . . , φk (znk )) ∈ X n+1 is admissible and nondegenerate when k is sufficiently large. t u

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5 Topological dimension In this section, we use the fact that barycentric simplices degenerate in dimensions greater than GeomDim(X) to show that the homological dimension of X and the topological dimension of compact subsets of X are both ≤ GeomDim(X). Lemma 5.1 Let X be a CBA space. 1. Any continuous map φ from a finite polyhedron P to X may be uniformly approximated by a Lipschitz map φ1 ; if GeomDim(X) ≤ n then we may arrange that φ1 (P ) has finite n-dimensional Hausdorff measure. 2. Suppose K ⊆ X is compact. For every > 0 the inclusion i : K → X can be -approximated by a Lipschitz map i1 : K → X which factors as f

g

K → P → X where P is a finite polyhedron, f, g are Lipschitz, and the f inverse image of each simplex of P has diameter < ; in particular the image of i1 has finite Hausdorff dimension. If K has zero k-dimensional Hausdorff measure then we may arrange that Dim(P ) < k; if GeomDim(X) = n < ∞ then we may arrange that Dim(P ) ≤ n. Proof. 1. Pick > 0. After barycentrically subdividing P enough times, we may assume that φ maps the vertices of each simplex τ of P to an admissible (j+1)-tuple in X (definition 4.1), and φ(τ ) ⊂ X has diameter < 2 . Then we may define φ1 : P → X to be the unique Lipschitz map whose restriction to each simplex of P is just the barycentric simplex determined by its vertices. Then d(φ, φ1 ) < , and if GeomDim(X) ≤ n then φ1 (P ) has finite ndimensional Hausdorff measure since it coincides with the image of the n-skeleton of P by Corollary 4.10. 2. Choose > 0 so that any tuple of points in K with pairwise distance < is admissible. Let U = {Bxi ( 2 )}i∈I be a finite open cover of K by open 2 -balls (in X) centered at points in K, let Y := ∪i∈I Bxi ( 2 ) ⊆ X, let N erve(U) be the simplicial complex whose simplices correspond to the subsets of U with nonempty intersection, and let |N erve(U)| be the polyhedron of N erve(U). Using barycentric simplices as in 1 above, we get a Lipschitz map σU from |N erve(U)| to X. By choosing a locally Lipschitz partition of unity {ρi }i∈I subordinate to U, we get a locally Lipschitz map ρU : Y → |N erve(U)|. The composition i1 := (σU ◦ ρU )|K satisfies the conditions in 2. If K has zero k-dimensional Hausdorff measure, then so does ρU (K) ⊆ |N erve(U)|, since ρU is Lipschitz on K. So we can define a Lipschitz map π from ρU (K) to the (k − 1)-skeleton of |N erve(U)| with the property that π(t) lies in the closed simplex determined by t for every t ∈ ρU (K) (starting with a top dimensional simplex τ of P , project away from a missed interior point to Bdy τ , etc). Now σU ◦ π ◦ ρU |K : K → K π◦ρ

σ

U X as desired. factors as K −→U |(N erve(U))k−1 | −→


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If GeomDim(X) ≤ n, then we already know that we can approximate iK : K → K with a Lipschitz map j1 whose image has finite n-dimensional Hausdorff measure, in particular zero (n + 1)-dimensional Hausdorff measure. So we may apply the previous paragraph to get a map j2 from j1 (K) to X which approximates the inclusion j1 (K) → X, and which factors through a polyhedron of dimension ≤ n; the composition j2 ◦ j1 is the desired approximation of ik . t u Proposition 5.2 Let X be a CBA space. Then 1. If k > GeomDim(X), then Hk (U, V ) = {0} for all open pairs (U, V ) in X. 2. sup{TopDim(Y ) | Y ⊆ X is compact} ≤ GeomDim(X). Proof. 1. Suppose k > n = GeomDim(X) and let (U, V ) be an open pair in X. Given [α] ∈ Hk (U, V ), there is a compact pair (K, L) and a map φ : (K, L) → (U, V ) so that [α] ∈ Im Hk (φ). Pick > 0. According to Lemma 5.1, we may approximate the inclusion φ(K) → X with maps i1 : φ(K) → X which factor through finite polyhedra of dimension ≤ n. If d(φ1 , φ) is sufficiently small then φ1 will be homotopic to φ (as a map of pairs) by the geodesic homotopy. In this case we have Im Hk (φ) = Im Hk (φ1 ) = {0}. 2. If K ⊆ X is compact, we may apply 2 of Lemma 5.1 to get a map f from K to a polyhedron P of dimension ≤ n so that Diam(f −1 (τ )) < for every simplex τ of P . This shows that open covers of K with order ≤ n + 1 are cofinal, so TopDim(K) ≤ n. t u Proposition 5.3 Let X be a CBA space with GeomDim(X) = n < ∞. Let I be a finite set, let {Ui }i∈I be open subsets of X such that all intersections Ui1 ∩ . . . ∩ Uik are either empty or contractible, and suppose ∪i∈I Ui ⊂ Y where Y is a contractible open set. Then if ∩i∈I Ui = ∅, there is a subset J ⊆ I with |J| ≤ n + 1 so that ∩i∈J Ui = ∅. Proof. Let k be the largest integer so that every subset I 0 ⊆ I with |I 0 | ≤ k gives ∩i∈I 0 Ui 6= ∅. Pick J ⊂ I with |J| = k + 1 so that ∩i∈J Ui = ∅, let V = {Ui }i∈J , and V = ∪i∈J Ui . The simplicial complex N erve(V) is isomorphic to Bdy(∆k ). As nonempty intersections are contractible, MayerVietoris sequences show that Hk−1 (V ) ' Hk−1 (Bdy(∆k )) 6= {0}. From the exact sequence of the pair (Y, V ) we get Hk (Y, V ) 6= {0}. By Corollary 5.2 we have k ≤ n. t u 6 Minimum sets and subsets homeomorphic to Rn In this section we study the images of barycentric simplices. The main results are Theorem 6.3, which produces biHölder images of open sets U ⊂ En

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in spaces with nondegenerate n-simplices, and Theorem 6.8, which gives biLipschitz images of open sets U ⊂ En in CBA spaces with geometric dimension n. 6.1 General minimum sets If X is a geodesic metric space and f = (f0 , . . . , fn ) : X → Rn+1 is a map with convex component functions, then f (X) ⊂ Rn+1 is not necessarily convex, but the “bottom” of f (X) behaves like a convex set in many respects. We will use this idea to produce nice subsets of X. Notation. If t, t0 ∈ Rn , we say that t ≺ t0 (resp. t t0 ) if ti < t0i (resp ti ≤ t0i ) for all 1 ≤ i ≤ n. CH(Y) denotes the convex hull of Y ⊂ Rn . Definition 6.1 If S ⊂ Rn , then an element s ∈ S ⊂ Rn is minimal if there is no s0 ∈ S with s0 ≺ s. Minset(S) denotes the set of minimal elements of S; M inset(S) is a closed subset of S. If J ⊂ {1, . . . , n}, let πJ : Rn −→ R|J| be the corresponding projection. The J-face of S is πJ−1 (M inset(πJ (S))). s ∈ M inset(S) is nondegenerate if it does not lie in the J-face of S for any nonempty proper subset J ⊂ {1, . . . , n}; Nondeg(S) denotes the set of nondegenerate points of S. Let X be a bounded complete length space with unique geodesic segments joining pairs of points. Given x0 , x1 ∈ X and λ ∈ [0, 1], we let (1 − λ)x0 + λx1 denote γ(λ) where γ : [0, 1] → X is the unique constant speed geodesic joining x0 to x1 . Definition 6.2 Suppose C > 0, and let f0 , . . . , fn be L-Lipschitz C-convex nonnegative functions on X. Set f := (f0 , . . . , fn ) : X −→ Rn+1 , := {t ∈ Rn+1 | ti ≥ 0 for all i}, M inset(f ) := S := f (X) ⊂ Rn+1 ≥ f −1 (M inset(S)), N ondeg(f ) := f −1 (N ondeg(S)). Theorem 6.3 1. If N > 1,P s1 , . . . , sN ∈ S are distinct elements, λk > 0 0 for 1P ≤ k ≤ N , and N k=1 λk = 1, then there is an s ∈ S with 0 s ≺ λk sk . ¯ = M inset(S). 2. M inset(S) 3. If x ∈ M inset(f ) then there P is a system of weights α ∈ ∆n := {t ∈ n+1 | t ≥ 0 for all i and R ti = P 1} so that x is a strict minimum for i P αi fi . Moreover, setting φ(t) := αi ti , s = f (x), we have φ(s − s0 ) ≥

C k s − s0 k2l∞ 2L2

(6.4)

for every s0 ∈ S, where k · kl∞ is the l∞ norm on Rn+1 . Consequently M inset(f ) is precisely the image of the barycentric simplex σf : ∆n −→ X, and N ondeg(f ) is nonempty iff σf is nondegenerate.


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4. If CH(S) is the convex hullPof S and p ∈ Bdy(CH(S)) admits a supporting linear functional ni=0 αi ti with αi > 0 for all 0 ≤ i ≤ n, then p ∈ M inset(S). 5. f |M inset(f ) : M inset(f ) −→ Rn+1 is a Lipschitz homeomorphism

onto M inset(S) with 12 -Hölder inverse. The image f (M inset(f )) = M inset(S) is a Lipschitz graph over a closed subset of the hyperplane P H = {t ∈ Rn+1 | ti = 0}. N ondeg(S) projects to an open subset of H, so N ondeg(f ) is biHölder to an open subset of Rn .

Corollary 6.5 If X is a CBA space with GeomDim(X) ≥ n, then there is an open set U ⊂ En , and a 12 -Hölder embedding U → X with Lipschitz inverse. Proof of Corollary 6.5. By Lemma 4.11 there is an admissible z ∈ X n+1 so that the barycentric simplex σz is nondegenerate. Setting f = (f0 , . . . , fn ) = (d2z0 , . . . , d2zn ) we see by 3 of Theorem 6.3 that N ondeg(f ) is nonempty; hence 5 of Theorem 6.3 applies. t u P PN λk Proof. 1. Since N k=1 λk sk = λ1 s1 + (1 − λ1 )( k=2 (1−λ1 ) sk ), 1 follows from the strict convexity of the fi ’s and induction. ¯ and pick a sequence xk ∈ X so 2. Let s ∈ S¯ be minimal in S, that f (xk ) → s. By the argument of Lemma 2.3, xk is Cauchy and so f (limk→∞ xk ) = s, which proves 2. 3. Suppose x ∈ M inset(f ), s = f (x). s is minimal in CH(S), for otherwise by 1, s would not be minimal in S. Therefore the convex set n+1 |t ≺ s}, and CH(S) is disjoint from the open convex set P U = {t ∈ R n+1 ∗ ) \{0}, φ(t) = αi ti , so that inf φ(CH(S)) ≥ hence there is a φ ∈ (R sup φ(U ). So we have αi ≥ 0 and without loss of generality we may assume P αi = 1. φ is the supporting linear functional sought in 3. The C-convex function φ ◦ f attains a minimum at x, so for every x0 ∈ X we have (φ ◦ f )(x0 ) ≥ (φ ◦ f )(x) +

C 2 d (x, x0 ). 2

If s0 = f (x0 ) ∈ S, then k s − s0 kl∞ ≤ Ld(x, x0 ), so the inequality (6.4) follows. functional Pn4. Suppose p ∈ Bdy(CH(S)) is supported by a linear n+1 α t with α > 0. Since S is a bounded subset of R , every i i i i=0 P p ∈ CH(S) is a convexP combination N µ s ¯ of at most n + 2 elek=1 k k ¯ ¯ µk = 1. If p 6∈ S then N ≥ 2, and uniform ments s¯k ∈ S, µk > 0, convexity of the fi gives us s0 ∈ SP with s0 ≺ p, contradicting the assumption that CH(S) is supported by αi ti at p. Hence p ∈ S¯ and clearly ¯ so by 2 we have p ∈ S, proving 4. p ∈ M inset(S)

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5. Pick x1 , x2 ∈ M inset(f ), and set d = d(x1 , x2 ), d0 = d(f (x1 ), f (x2 )). For each i we have fi

1 1 x1 + x2 2 2

1 1 C ≤ fi (x1 ) + fi (x2 ) − 2 2 2

2 d 2

by the C-convexity of fi , and 1 1 d0 fi (x1 ) + fi (x2 ) − fi (x1 ) ≤ 2 2 2 by the triangle inequality. Hence fi

1 1 x1 + x2 2 2

C + 2

2 d d0 − fi (x1 ) ≤ . 2 2

Since x1 ∈ M inset(f ), we have fi ( 12 x1 + 12 x2 ) ≥ fi (x1 ) for some 0 ≤ 2 older i ≤ n, so d0 ≥ Cd 4 . This shows that f |M inset(f ) satisfies a reverse H¨ inequality, and so f has a Hölder inverse. This proves the first claim of 5. To see that M inset(S) ⊂ Rn+1 is a Lipschitz P graph over a closed subset of the hyperplane H := {t ∈ Rn+1 | i ti = 1}, note that if p ∈ M inset(S) then M inset(S) is disjoint from the union of the orthants {t ∈ Rn+1 | t ≺ p} and {t ∈ Rn+1 | p ≺ t}. Now suppose p ∈ N ondeg(S). By 3, p lies on the boundary of CH(S) P so CH(S) can be supported at p by a linear functional φ(t) := αi ti , P 0 0 αi ≥ 0. If φ (t) = αi ti is any supporting functional for CH(S) at p, φ and φ0 would then αi0 > 0 for all i; otherwise a convex combination P of 00 00 αi ti where αi00 ≥ 0 give another supporting linear functional φ (t) = 00 for all i and αi = 0 for some i, contradicting the nondegeneracy of p. By an obvious limiting argument, any point p0 ∈ Bdy(CH(S)) sufficiently close to p also has the property that every supporting linear functional has positive coefficients; consequently, a neighborhood U of p in Bdy(CH(S)) is a convex hypersurface contained in N ondeg(S) by 4, and this neighborhood projects to an open set in H. t u 6.2 Top dimensional minimum sets We now specialize the setup of Sect. 6.1 to the case when X is a CAT (κ) space with the property that d(x1 , x2 ) is less than D(κ) (the diameter of the model space Mκ2 with constant curvature κ) for all x1 , x2 ∈ X, the C-convex functions f0 , . . . , fn are the squares of distance functions dzi , zi ∈ X, and GeomDim(X) = n.


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Theorem 6.6 If K is a compact subset of N ondeg(f ), then there is a constant L1 > 0 so that for every x ∈ K and every x0 ∈ X we have d(f (x), f (x0 )) ≥ L1 d(x, x0 )

(6.7)

In particular, f |K : K −→ Rn+1 is a biLipschitz homeomorphism onto its image. By part 5 of Theorem 6.3, N ondeg(S) is the graph (over the hyperplane H) of a Lipschitz function defined on an open subset of H. Therefore N ondeg(f ) is locally biLipschitz to an open subset of En . Proof. If x ∈ N ondeg(f ) and 0 ≤ i ≤ n, then there is a y ∈ X so that ∠x (y, zj ) < π2 for every j 6= i. Upper semicontinuity of the angle function x 7→ ∠x (y, zi ) (see Sect. 2.1) implies that there is an rx < π2 so that ∠x0 (y, zj ) ≤ rx for every j 6= i when x0 is sufficiently close to x. Hence by the compactness of K, we may conclude that the points in K are “uniformly nondegenerate” in the sense that there is an r < π2 so that for →) ≤ r xz every x ∈ K and every 0 ≤ i ≤ n there is a v ∈ Σx X with ∠x (v, − j for j 6= i. We now show that the failure of a reverse Lipschitz condition (6.7) implies that GeomDim(Σx X) ≥ n for some x ∈ K, which contradicts our assumption that n = GeomDim(X) ≥ 1 + GeomDim(Σx X). Choose d(f (x ),f (x0 )) sequences xk ∈ K, x0k ∈ X − {xk } so that limk→∞ d(xkk ,x0 ) k = k 0. Note that limk→∞ d(xk , x0k ) = 0, for otherwise the C-convexity of the fi ’s implies that the midpoint mk of the segment xk x0k will satisfy f (mk ) ≺ f (xk ) for sufficiently large k, contradicting xk ∈ M inset(f ). e x (x0 , zi ) satisfy limk→∞ ∠ e x (x0 , zi ) = π , Hence the comparison angles ∠ k k k k 2 π 0 so lim supk→∞ ∠xk (xk , zi ) ≤ 2 . Applying the preceding paragraph, for → each 0 ≤ i ≤ n and every k choose vki ∈ Σxk X such that ∠xk (vki , − x− k zj ) ≤ π → (r) r < 2 for j 6= i. Hence for sufficiently large k, we have that ∩0≤i≤n B− x− k zi π → ( ) = ∅, ∩j6=i Bx −−→ → (r) (r) 6= ∅ for every i, and ∪0≤i≤n Bx−− ⊆ ∩0≤i≤n Bx−− k zi 2 k zj k zi ⊆ B−−→ (π). By Proposition 5.3 we have GeomDim(Σ X) ≥ n, which xk x x0 k k

is a contradiction.

t u

Let X, zi , fi be as in the previous theorem. Theorem 6.8 Suppose x0 ∈ N ondeg(f ), and let s0 = f (x0 ) be a point where the convex P hypersurface f (N ondeg(f )) = N ondeg(S) is differentiable. Let φ P = αi ti be the supporting linear functional for M inset(S) at s0 , αi > 0, αi = 1, let P = φ−1 (φ(x0 )) be the supporting hyperplane, −1 : M inset(S) −→ Q := Cs0 P ⊂ Cs0 Rn+1 , and g = f |M inset(f ) M inset(f ). Then

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1. g has a well-defined tangent cone Cs0 g : Cs0 P = Cs0 M inset(S) = Q −→ Cx0 X, in the sense that if xk ∈ M inset(f ), sk = f (xk ) ∈ M inset(S) \ {s0 }, limk→∞ sk = s0 , and d(sk1,s0 ) logs0 sk ∈ Cs0 Rk+1 converges to v ∈ Q, then d(sk1,s0 ) logx0 (xk ) converges to (Cs0 g)(v). ˆ = Im(Cs g) at Hence M inset(f ) has a well-defined tangent cone Q 0 n x0 which is isometric to E . Cs0 g is an affine map. 2. With respect to a suitable Euclidean metric d1 on Rn+1 , we find that lim

s1 ,s2 →s0

d(g(s1 ), g(s2 )) = 1. d1 (s1 , s2 )

In particular, for every > 0 we have an open subset U ⊂ En and a (1 + )-biLipschitz embedding U −→ X. → ˆ 3. − x− 0 zi ∈ Q. We remark that the top dimensionality (the number of functions is 1 + GeomDim(X)) of the nondegenerate minimum set is essential in Theorem 6.8, see example 6.14. Proof. The idea of the proof is that near s0 , N ondeg(S) can be viewed as a graph over P of a function with small Lipschitz constant; this together with Theorem 6.6 forces f |M inset(f ) to be “approximately affine” and M inset(f ) to be “approximately convex” near x0 . Lemma 6.9 (f is approximately affine) There is a function (r) with limr→0 (r) = 0 with the following property. If x1 , x2 ∈ M inset(f ), si = f (xi ) ∈ M inset(S); d(xi , x0 ) < r; λ1 , λ2 are weights; s3 ∈ M inset(S) satisfies s3 λ1 s1 + λ2 s2 ; f (x3 ) = s3 ; then d(x3 , λ1 x1 + λ2 x2 ) ≤ (r)d(x1 , x2 )M in(λ1 , λ2 )

(6.10)

and 1 − (r) ≤

d(x1 , x3 ) d(x2 , x3 ) ≤ 1 + (r), 1 − (r) ≤ ≤ 1 + (r). λ2 d(x1 , x2 ) λ1 d(x1 , x2 ) (6.11)

In particular, the comparison triangles for ∆(x1 , x2 , x3 ) and ∆(s1 , s2 , s3 ) have angles tending to {0, π} as r → 0. Proof. Since N ondeg(S) is (locally) a convex hypersurface and s0 ∈ N ondeg(S) is a differentiable point, the supporting hyperplanes at points


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s ∈ N ondeg(S) tend to P as s → s0 . Therefore if πP : Rn+1 −→ P is the orthogonal projection, we have lim

d(si ,s0 )→0

d(s1 , s2 ) =1 d(πP (s1 ), πP (s2 ))

and N ondeg(S) ∩ Bs0 (r) is the graph over P of a function with Lipschitz constant tending to zero with r. Hence if λ1 , λ2 are weights, s1 , s2 , s3 ∈ N ondeg(S), si ∈ Bs0 (r) and s3 λ1 s1 + λ2 s2 , we have d(s3 , λ1 s1 + λ2 s2 ) ≤ 1 (r)d(s1 , s2 )M in(λ1 , λ2 ) where limr→0 1 (r) = 0. With xi = g(si ) we get L1 d(x3 , λ1 x1 + λ2 x2 ) ≤ d(f (x3 ), f (λ1 x1 + λ2 x2 )) = d(s3 , f (λ1 x1 + λ2 x2 )) by Theorem 6.6, but since f (λ1 x1 + λ2 x2 ) λ1 s1 + λ2 s2 we have d(s3 , f (λ1 x1 + λ2 x2 )) ≤ 21 (r)d(s1 , s2 )M in(λ1 , λ2 ). (6.10) now follows from the fact that f is Lipschitz. (6.11) follows from (6.10) (after adjusting (r) if necessary) and the triangle inequality. t u It follows easily from (6.10) and (6.11) that if sk ∈ M inset(S) \ {s0 } is a sequence with limk→∞ sk = s0 , and d(sk1,s0 ) logs0 sk ∈ Cs0 Rn+1

0 ,xk ) converges to some v ∈ Q, then setting xk = g(sk ), limk→∞ d(x d(s0 ,sk ) and limk→∞ d(x01,xk ) logx0 xk exist. We therefore have a well-defined map Cs0 g : Q −→ Cs0 X. The estimates (6.10) and (6.11) also imply that Cs0 g is an affine map in the sense that for every v1 , v2 ∈ Q we have (Cs0 g)(λ1 v1 + λ2 v2 ) = λ1 (Cs0 g)(v1 ) + λ2 (Cs0 g)(v2 ) for any weights ˆ := (Cs g)(Q) is a convex subset. Also, the fact that g is λ1 , λ2 . Hence Q 0 locally biLipschitz near s0 (Theorem 6.6) implies that Cs0 g is biLipschitz. If γ1 , γ2 : R −→ Q are two constant speed geodesics with d(γ1 (t), γ2 (t)) < C, then η1 = (Cs0 g) ◦ γ1 and η2 = (Cs0 g) ◦ γ2 are constant speed ˆ which satisfy d(η1 (t), η2 (t)) ≤ L1 C geodesics in the Hadamard space Q −1 since g is L1 -Lipschitz; therefore d(η1 (t), η2 (t)) is constant and the ηj ’s ˆ is flat, and we have proved 1 of bound a flat strip. Hence it follows that Q Theorem 6.8. ˆ by We now prove 2. Let h·, ·iQ be the inner product on Q induced from Q n+1 which extends h·, ·iQ and Cs0 g; let h·, ·i1 be the inner product on Cs0 R which defines the same Q⊥ as the standard inner product, and which agrees with the standard inner product on Q⊥ . Let d1 be the distance function on Rn+1 ' Cs0 Rn+1 defined by h·, ·i1 . We now have that

lim

s1 →s0

d(g(s1 ), g(s0 )) d1 (s1 , s0 )

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= lim

s1 →s0

d(g(s1 ), g(s0 )) d(s1 , s0 ) d(s1 , s0 ) d1 (s1 , s0 )

= 1.

This implies the third assertion of theorem 6.8 since if s1 , s2 ∈ M inset(S) are sufficiently close to s0 , there will be an s3 ∈ M inset(S) with d(s3 , s0 ) e s (s2 , s3 ) 1; so by (6.10) and (6.11) max(d(s1 , s0 ), d(s2 , s0 )) and with ∠ 1 we get d(g(s1 ), g(s2 )) 1. − 1 d1 (s1 , s0 ) It remains to prove 3. Lemma 6.12 For 0 ≤ i ≤ n let Dfi : Cx0 X −→ R denote P the derivative of → fi , Dfi (w) = −2d(x0 , zi )hw, − x− αi Dfi is zero on 0 zi i (cf Sect. 2.2). Then ˆ where φ(t) = P αi ti is the supporting linear functional for M inset(Q) Q, ˆ for each i. at s0 . Consequently Dfi is affine on Q ˆ w = (Cs g)(v), then Proof. If w ∈ Q, 0 X X αi Dfi (v) = αi Dfi ((Cs0 g)(v)) = Dφ(v) = 0. t u → ˆ Lemma 6.13 For 0 ≤ i ≤ n, − x− 0 zi ∈ Q. ˆ implies that if w1 , w2 ∈ Q ˆ ∩ Σx X Proof. The affineness of the Dfi on Q 0 − − → − → ˆ are opposite directions in the flat Q, then ∠(w1 , x0 zj ) + ∠(w2 , x− 0 zj ) = π. −−→ −−→ ˆ ˆ Therefore if x 0 zj 6∈ Q, we have max{∠(w, x0 zj ) | w ∈ Q ∩ Σx0 X} < π, ˆ → (π). By so the (n − 1) sphere Q ∩ Σx0 X lies in the contractible set B− x− 0 zj Proposition 5.3 this implies that GeomDim(Σx0 X) ≥ n, which contradicts our assumption that GeomDim(X) = n. Hence the lemma. t u This completes the proof of Theorem 6.8

t u

Example 6.14 This example shows that the top dimensionality assumption of Theorem 6.8 is necessary. If z ∈ X n+1 , fi := d2zi and p ∈ M inset(f ) is a nondegenerate point so that f (p) is a differentiable point of f (M inset(f )), then Cp X need not contain an n-flat. Start with the standard upper hemi2 ⊂ S 2 , and pick equally spaced points ξ , ξ , ξ ∈ ∂S 2 . Let sphere S+ 0 1 2 + 2 2 , let U be B ⊂ S+ be a spherical cap centered at the North pole N ∈ S+ i the geodesic cone over B with vertex at ξi , and set U = ∪Ui . Modify the 2 \ U so that it has curvature K < 1, and so that it is in polar metric on S+ coordinate form with radius π2 . Then we have a CAT (1) space Y , and let X be the Euclidean cone over Y with vertex o. Now if we take zi ∈ X to lie on the ray oξi , then the vertex o ∈ Co X will correspond to a differentiable point of dz (M inset(z)), but Cp X doesn’t contain any flats.


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7 The Tits boundary, asymptotic cones, flats, and the geometric dimension For properties of geometric boundaries, Tits boundaries, ultralimits, and asymptotic cones, see Sect. 2 or [KL97]. In this section ω will be a fixed non-principal ultrafilter on N. Theorem 7.1 Let {(Xi , di )}i∈I be a family of Hadamard spaces. Then the following are equivalent: 1. There are sequences ik ∈ I, ?k ∈ Xik , ?k ∈ Fk ⊂ Xik , so that (Fk , ?k ) converges to (Er , 0) in the pointed Gromov-Hausdorff topology. Equivalently, ω-lim(Xik , ?k ) contains an r-flat. 2. There are sequences ik ∈ I, Rk ∈ R, Sk ⊂ Xik , so that limk→∞ Rk = ∞, and R1k Sk converges to B(1) ⊂ Er in the Gromov-Hausdorff topology. Equivalently, ω-lim( R1k Xik , ?k ) contains an isometric copy of B(1). 3. There are sequences ik ∈ I, ?k ∈ Xik , λk → ∞ so that ω-lim( λ1k Xik , ?k ) is a CAT (0) space with geometric dimension ≥ r. 4. There are sequences ik ∈ I, ?k ∈ Xik , so that ∂T (ω-lim(Xik , ?k )) has geometric dimension ≥ r − 1. Remark. In 2, we really only use the fact that the R1k Sk contain subsets which converge to {±e1 , . . . , ±er , 0} ⊂ Er , where ei is the ith standard basis vector. Corollary 7.2 If the Xi ’s have uniformly bounded geometry (in the sense that {(Xi , ?) | i ∈ I, ? ∈ Xi } is a Gromov-Hausdorff precompact family of pointed metric spaces), then there is an r0 ∈ N such that for any sequences λk ∈ R, ik ∈ I, ?k ∈ Xik with ω-lim λk = 0 we have 1. GeomDim(ω-lim( λ1k Xik , ?k )) ≤ r0 . 2. GeomDim(∂T (ω-lim(Xik , ?k ))) ≤ r0 . Proof of Corollary 7.2. The uniformly bounded geometry of the Xi ’s implies that there is an r0 ∈ N so that for any sequence ik ∈ I and any sequence of basepoints ?k ∈ Xik , ω-lim(Xik , ?k ) contains no r-flats with r > r0 . Therefore the corollary follows from Theorem 7.1. t u Proof of Theorem 7.1. Clearly 1 implies 2 , 3 , and 4. 3 =⇒ 2. Let Xω = ω-lim( λ1k Xik , ?k ). Since GeomDim(Xω ) ≥ r we can find p ∈ Xω so that GeomDim(Σp X) ≥ r − 1. By Lemma 3.1, there are sequences T¯j ⊂ Xω , Rj ∈ R so that limk→∞ d(p, T¯j ) → 0, and 1 T¯j Rj

converges to B(1) ⊂ Er in the Gromov-Hausdorff topology. This means that for each j we can find sequences i0k ∈ N, λ0k ∈ R, Tkj ⊂ Xi0k so that 10 T j converges to T¯j in the Gromov-Hausdorff topology. Passing to a λk

k

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suitable subsequence of the double sequence Tkj we get a sequence Sk as described in 2. 4 =⇒ 2. Let Y be the Euclidean cone over ∂T (ω-lim(Xik , ?k )), with vertex o. By Lemma 3.1, there are subsets Tk ⊂ Y and a sequence λk → 0 so that λ1k Tk converges to B(1) ⊂ Er in the Gromov-Hausdorff topology, and d(Tk , o) → 0. Since any finite metric space Z ⊂ Y is a Hausdorff limit of rescaled finite metric spaces in Xi (this follows easily from the definition ¯ j → ∞, T¯j ⊆ ω-lim(Xi , ?k ) so that of the Tits metric), we get sequences λ k 1 ¯ r ¯ ¯ Tj converges to B(1) ⊂ E in the Gromov-Hausdorff topology. Each Tj λj

is the Gromov-Hausdorff limit of a suitable sequence of elements of {Si }i∈I so 2 follows. 2 =⇒ 1. We will show that for a suitable choice of basepoints ?k ∈ Xik the ultralimit ω-lim(Xik , ?k ) contains an r-flat. To simplify notation slightly we assume that I = N and that ik = k ∈ N. We will also assume that r ≥ 2 since the implication is trivial otherwise. By assumption Xk ⊃ R1k Sk −→ B(1) ⊂ Er in the Gromov-Hausdorff topology, so we can find a sequence φk : Er ⊃ B(0, Rk ) → Xk of k Rk Hausdorff approximations (definition 2.6) where k → 0. For 1 ≤ i ≤ r let ei ∈ Er be the ith standard basis vector, let zk±i := φk (±Rk ei ) and let Zk := {zk±i }1≤i≤r ⊂ Xk . Define fk : Xk → R to be the average 1 P distance from Zk ⊂ Xk , i.e. fk (p) := 2r z∈Zk d(p, z). fk is 1-Lipschitz and convex. Choose ?k ∈ Xk so that fk (?k ) ≤ k +inf Xk fk . We will extract an ultralimit of the configuration {?k zk±i }1≤i≤r of geodesic segments to get a configuration of geodesic rays and then we will show that these geodesic rays span an r-flat. Lemma 7.3 limk→∞

1 Rk d(?k , φk (0))

Proof. We have lim supk→∞ On the other hand lim inf k→∞

= 0.

1 Rk fk (?k )

≤ lim supk→∞

1 Rk fk (φk (0))

= 1.

1 1 [d(zki , ?k ) + d(?k , zk−i )] ≥ lim inf d(zki , zk−i ) = 2, k→∞ Rk Rk

so lim inf k→∞ R1k fk (?k ) ≥ 1. Combining these inequalities we get limk→∞ R1k fk (?k ) = 1, limk→∞ R1k [d(zki , ?k ) + d(?k , zk−i )] = 2,

limk→∞ R1k d(?k , zki zk−i ) = limk→∞ R1k d(?k , φk (0)zki ∪ φk (0)zk−i ) = 0. This forces limk→∞ R1k d(?k , φk (0)) = 0 when r ≥ 2 since e φ (0) (z ±i , z ±j ) = π when i 6= j. limk→∞ ∠ u t k k k 2 Now consider the ultralimit (Xω , ?ω ) := ω-lim(Xk , ?k ). We have geodesic rays ω-lim ?k zk±i with ideal boundary points ξ ±i ∈ ∂T Xω . If we let


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fk±i (·) := d(zk±i , ·) − d(zk±i , ?k ) be the normalized distance function, then ω-lim fk±i is the Busemann function bξ±i of the geodesic ray ?ω ξ ±i . There1 P fore ?ω is a minimum of b := 2r i (bξ i +bξ −i ) because if xω = ω-lim xk ∈ Xω , then b(?ω ) = 0 = ω-lim(−k ) ≤ ω-lim[fk (xk ) − fk (?k )] 1 X i = ω-lim [fk (xk ) + fk−i (xk )] 2r i

1 X = [bξi (xω ) + bξ−i (xω )] =: b(xω ). 2r i

e e ? (z ±i , z ±j ) = π (∠ On the other hand, by Lemma 7.3 we have ω-lim ∠ k k k 2 denotes the comparison angle, see Sect. 2.1) when 1 ≤ i 6= j ≤ r, so we −−−→ conclude that ∠T (ξ ±i , ξ ±j ) ≤ π2 for 1 ≤ i 6= j ≤ r. Set w±i := ?ω ξ ±i ∈ Σ?ω Xω . We have ∠?ω (w±i , w±j ) ≤ ∠T (ξ ±i , ξ ±j ) ≤ π2 . The directional derivative of b in the direction of v ∈ Σ?ω Xω is (see equation 2.5) 1 X i − [hw , vi + hw−i , vi]. 2r i

With v = w±j we find that the directional derivative is ≤ 0 since hw±i , w±j i ≥ 0 when i 6= j and hwj , w±j i + hw−j , w±j i ≥ 0. As ?ω minimizes b we have equality everywhere, forcing ∠?ω (wi , w−i ) = ∠T (ξ i , ξ −i ) = π and ∠?ω (w±i , w±j ) = ∠T (ξ ±i , ξ ±j ) = π2 for i 6= j. This easily implies that the convex hull of the rays ?ω ξ ±i is an r-flat Fω ⊂ Xω (to see this, assume inductively that the convex hull CH({?ω ξ ±i }i≤m ) is an m-flat Fm in Xω , and observe that the nonnegative convex function bξ+(m+1) + bξ−(m+1) is zero on Fm , so the convex hull CH(Fm ∪ {?ω ξ ±(m+1) } is isometric to u t Fm × R ' Em+1 ). Corollary 7.4 If X is a locally compact CAT (0) space with cocompact isometry group, then every compact subset of any asymptotic cone of X has topological dimension ≤ TopDim(X). 8 Proofs of Theorems A, B, and C Proof of Theorem A. For 1 ≤ i ≤ 4 let ni ∈ N ∪ {∞} denote the ith quantity in the statement of Theorem A. Lemma 3.1 proves GeomDim(X) ≤ n3 . Proposition 5.2 proves n1 ≤ GeomDim(X) and n2 ≤ GeomDim(X); Proposition 3.2 proves n3 ≤ n1 and n3 ≤ n2 . Obviously n4 ≤ n1 . Theorem t u 6.3 proves GeomDim(X) ≤ n4 .

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Proof of Theorem B. Let n = GeomDim(X). By Lemma 4.11 there is a z ∈ X n+1 which is (κ, r)-admissible for some κ, r, such that σz is nondegenerate. By 3 of Theorem 6.3 if we let fi := d2zi : ∩Bzi (r) → R, then N ondeg(f ) is nonempty. By 1 of Theorem 6.8 we have p ∈ X whose space of directions contains an isometrically embedded standard (n − 1)sphere Z ⊆ Σp X. Since Z is an absolute neighborhood retract, there is an open neighborhood V ⊂ Σp X of Z so that Hn−1 (Z) −→ Hn−1 (V ) is a monomorphism. Then Hn−1 (V ) −→ Hn−1 (Σp X) is also a monomorphism since Hn (Σp X, V ) = 0 by Theorem A. So Hn−1 (Σp X) 6= 0. We may find a map φ : Z −→ X − {p} so that φ(Z) is in the domain Y of logΣp X and logΣp X ◦φ is arbitrarily close to the inclusion Z −→ Σp X. Identifying Z with the unit sphere S n−1 ⊂ En by an isometry, we define a map φ¯ : ¯ is the En ⊃ B n → Y by declaring that if λ ∈ [0, 1], x ∈ S n−1 , then φ(λx) unique point on the segment pφ(x) at distance λd(p, φ(x)) from p. Hence we get a map of pairs φ¯ : (B n , S n−1 ) −→ (Y, Y − {p}) which is nontrivial on Hn since its boundary is nontrivial. Hence Hn (X, X − {p}) 6= {0} by excision. This shows that each of the quantities in the statement of Theorem B is ≥ GeomDim(X). The remaining inequalities are contained in Theorem A. t u Proof of Theorem C. See Theorem 7.1.

t u

9 Questions from Asymptotic Invariants of infinite groups On pp.127-33 of [Gro93] Gromov discusses a number of issues relating to the large-scale geometry of Hadamard spaces. The main results of this paper – especially the more general version of Theorem C formulated in Theorem 7.1 – settle many of the questions raised in Gromov’s discussion provided one replaces topological dimension (Gromov’s “dim”) with compact topological dimension: Definition 9.1 If Z is a topological space, then the compact topological dimension of Z is CTopDim(Z) := sup{TopDim(K) | K ⊂ Z is compact}. We now comment on some of the questions. DimConω X ≤ DimX, [Gro93, p.129]. We reformulate this as follows: if X is an arbitrary Hadamard space, then CTopDim(Xω ) ≤ CTopDim(X) for any asymptotic cone of X. To see this, suppose Xω is an asymptotic cone of X and CTopDim(Xω ) ≥ k. Then by Theorem A we have sequences Rj → 0, Sj ⊂ Xω so that d(p, Sj ) → 0 for some p ∈ Xω and R1j Sj → B(1) ⊂ Ek in the Gromov-Hausdorff topology. From the properties of


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ultralimits (see Sect. 2.3 property 5) we see that there are sequences Rj0 → 0, Tj ⊂ X so that R10 Tj → B(1) ⊂ Ek in the Gromov-Hausdorff topology. j

By the remark after Proposition 3.2 we get CTopDim(X) ≥ k.

Existence of regular points, [Gro93, p.132-3]. Gromov defines a point p in a CBA space to be regular if the tangent cone Cp X satisfies DimCp X ≥ DimX, and asks when regular points exist. If we replace Dim with CT opDim, then Theorem A gives CTopDim(X) = GeomDim(X) = sup GeomDim(Cp X) p∈X

= sup CTopDim(Cp X). p∈X

So regular points always exist when CTopDim(X) < ∞. In the general case CTopDim(X) can be locally finite even when CTopDim(X) = ∞; for example, take the disjoint union E1 ∪ E2 ∪ . . . , and glue on a segment of length 1 starting at 0 ∈ E1 and ending at 0 ∈ Ek . Rank’s and Rank + ’s, [Gro93, pp.127-33]. Gromov gives seven definitions of rank for Hadamard spaces and then asks whether they coincide for Hadamard spaces with cocompact isometry groups, or, more generally, if the “plusified” ranks RankI+ , . . . , RankV+II agree for arbitrary Hadamard spaces. Theorem 7.1 in this paper settles this question completely for RankI+ , + provided one uses compact topological dimension instead of . . . , RankIV the usual topological dimension. Let (X, d) be a Hadamard space. To see that the ranks RankI+ , . . . , + (redefined using CTopDim instead of TopDim) are equal we will RankIV apply Theorem 7.1 to the one-element family of spaces (X, d). For 1 ≤ i ≤ 4 let ri ∈ N ∪ ∞ be the supremum of the r’s which satisfy the ith statement in Theorem 7.1. Let X + be the plusification of X, i.e. the collection of ultralimits of the form ω-lim(X, d, ?i ) where ?i is a sequence in X. + + Notice that r1 is exactly the same as RankII X. Similarly, RankIV X = r4 since by Theorem A, part 1 we have GeomDim(∂T Z) = CTopDim (∂T Z) for any Hadamard space Z. Suppose RankI+ X ≥ r. Then there is an X 0 ∈ X + and an asymptotic cone (X 0 )ω of X 0 with CTopDim((X 0 )ω ) ≥ r. Therefore by Theorem A, there are sequences Rk → 0, Sk ⊂ (X 0 )ω so that Sk is finite, and 1 r Rk Sk converges to B(1) ⊂ E in the Gromov-Hausdorff topology. Each Sk is a Gromov-Hausdorff limit of a sequence R1kl Skl where Skl ⊂ X 0 is finite and Rkl → ∞ (see Sect. 2.3 property 5); and each Skl ⊂ X 0 is a Gromov-Hausdorff limit of a sequence of finite subsets of X itself. By a diagonal construction we find sequences Rk0 → ∞ and Tk ⊂ X so

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that

B. Kleiner 1 T Rk0 k

→ B(1) ⊂ Er in the Gromov-Hausdorff topology. Hence 2 of

Theorem 7.1 is satisfied for this r. Taking suprema we get RankI+ X ≤ r2 . + Note that RankIII X ≤ RankI+ X since an r-quasiflat in X 0 ∈ X + produces a biLipschitz embedded copy of Er in asymptotic cones of X 0 . + + But r1 = RankII X ≤ RankIII X ≤ RankI+ X ≤ r2 so the Rank + ’s are all equal to the ri ’s. 10 Convex length spaces A convex (length) space is a complete length space (X, d) such that if γ1 : [a1 , b1 ] → X and γ2 : [a2 , b2 ] → X are constant speed geodesics, then the function d ◦ (γ1 , γ2 ) : [a1 , b1 ] × [a2 , b2 ] → R is convex. Convex spaces were introduced by Busemann (see [Bus55]); the literature about them seems to be quite limited: [Rin61, Gro78,Bow95,AB90]. Hadamard spaces are convex length spaces, as are Banach spaces with a strictly convex norm. Our goal in this section is to prove Theorem 1. In Sect. 10.1 we discuss elementary properties of convex spaces, in Sect. 10.2 we give a proof of a special case of the differentiation theorem from [KS93], and in Sect. 10.3 we construct flats in convex spaces whose asymptotic cone contains a flat. 10.1 Background We recall some basic facts about convex spaces, omitting proofs whenever the standard proofs in the Hadamard space case extend without significant modification to convex length spaces. Let X be a convex space. Two geodesic rays7 γ1 : [0, ∞) → X, γ2 : [0, ∞) → X are asymptotic if d(γ1 (t), γ2 (t)) is bounded. Given any p ∈ X and any geodesic ray γ1 : [0, ∞) → X there is a unique geodesic ray γ2 with γ2 (0) = p which is asymptotic to γ1 . Asymptoticity is an equivalence relation on geodesic rays, and we use C∞ X to denote the set of equivalence classes. For any p ∈ X we may view C∞ X as a subset of {γ ∈ C([0, ∞), X) | γ(0) = p}; the compact-open topology on C([0, ∞), X) induces a subspace topology on C∞ X, which is independent of p. The geometric boundary, ∂∞ X, is the subset of C∞ X determined by the unit ¯ := X ∪ ∂∞ X inherits a natural topology, speed geodesic rays. The union X which is compact when X is locally compact. The Tits distance between two geodesic rays γ1 , γ2 is dT its (γ1 , γ2 ) := lim

t→∞

d(γ1 (t), γ2 (t)) . t

7 In this section all geodesics rays will be parametrized at constant (not necessarily unit) speed.


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dT its defines a metric on C∞ X; we call the resulting metric space the Tits cone and denote it by CT X. dT its is lower semicontinuous with respect to the product topology on C∞ X × C∞ X. The Tits boundary is the subset ∂T X ⊂ CT X determined by the unit speed geodesic rays. If p ∈ X, and ξ ∈ ∂T X, then pξ denotes (the image of) the geodesic ray with initial point p. We define Busemann functions for unit speed geodesic rays as in the Hadamard case; unlike the Hadamard case, the Busemann functions of asymptotic geodesic rays need not differ by a constant. However, the following properties still hold: Lemma 10.1 1. If pk , xk ∈ X, and the geodesic segments pk xk converge to a geodesic ray p∞ ξ ⊂ X, then the Busemann function bξ of the ray p∞ ξ satisfies bξ ≥ lim sup [d(xk , ·) − d(xk , pk )] k→∞

(10.2)

Recall that in the Hadamard case bξ is the limit of the normalized distance functions [d(xk , ·) − d(xk , pk )]. 2. If p ∈ X, ξ ∈ ∂T X, bξ is the Busemann function for the ray pξ, and the unit speed geodesic ray γ : [0, ∞) → X is a “gradient line” for bξ : bξ (γ(t)) = bξ (γ(0)) − t, then γ is asymptotic to pξ. 3. If p ∈ X, ξ1 , ξ2 ∈ ∂T X, bξi denotes the Busemann function of pξi , and pξ1 ∪ pξ2 forms a geodesic, then bξ1 + bξ2 ≥ 0 on X. In this case if x ∈ X and (bξ1 + bξ2 )(x) = 0, then xξ1 ∪ xξ2 is a geodesic parallel to pξ1 ∪ pξ2 . We will use the following result: Theorem 10.3 ([Rin61, p.432, par. 7 and p. 463, par.20], [Bow95, Lemma 1.1 and remark after its proof]) If X is a convex length space, and γ1 , γ2 : R → X are constant speed geodesics with d(γ1 (t), γ2 (t)) < C, then γ1 and γ2 bound a flat Minkowskian strip (a convex subset isometric to the region in a normed plane bounded by two parallel lines.). A weaker convexity condition. Convexity is not preserved by limit operations: a sequence of strictly convex norms on Rn may converge to a nonstrictly convex norm. To remedy this defect we introduce a weaker convexity condition below: we only insist on the convexity of the distance function when it is restricted to a distinguished collection of geodesic segments. Definition 10.4 A family G of constant speed geodesics (or geodesic segments) in a length space X is adequate if

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a) Each pair of points in X is joined by a geodesic segment in G; and b) G is closed under precomposition with affine maps: if γ ∈ G, γ : [a, b] → X, and α : [c, d] → [a, b] is affine, then γ ◦ α : [c, d] → X is in G. Definition 10.5 A length space X with distance function d is convex with respect to an adequate family of geodesics G in X if d ◦ (γ1 , γ2 ) is convex for all pairs γ1 , γ2 ∈ G. X is often convex if it is convex with respect to some adequate family of geodesics. If for each i ∈ N, Xi is a length space convex with respect to Gi , then for any choice of basepoints ?i ∈ Xi , the ultralimit (Xi , ?i ) is convex with respect to the adequate family Gω := {ω-lim γi | ∃C > 0 such that ∀i ∈ N, γi : [a, b] → Xi , d(γi (a), ?i ), d(γi (b), ?i ) < C}. In particular, any asymptotic cone of an often convex length space is an often convex length space. Lemma 10.6 (B. Leeb) Let (X, d) be a locally compact convex length space. If λi → 0, and (Xω , dω , ?ω ) := ω-lim(X, λi d, ?) (i.e. Xω is an asymptotic cone with fixed basepoints), then there is a canonical isometric embedding i : CT X → Xω and a 1-Lipschitz retraction ρ : Xω → i(CT X). Proof. Given a geodesic ray γ : [0, ∞) → X, (γ( λ1i )) defines a point i(γ) in Xω . This clearly defines an isometric embedding i : CT X → Xω . To ¯ := X ∪ ∂∞ X and let obtain the retraction ρ, we use the compactification X ¯ normalized by ω-lim λi d(xi , ?). ρ((xi )) ∈ CT X be the ultralimit of xi ∈ X t u 10.2 Differentiating maps into metric spaces The next result is a slight reformulation of a special case of the differentiation theory of [KS93, Sect. 1.9]. Since this case is somewhat simpler than the general Lp version in [KS93], we give a proof here. We will use d0 , k · k0 , and B0 (x, r) to denote the Euclidean metric, the Euclidean norm, and a Euclidean ball, respectively. Theorem 10.7 (Korevaar-Schoen) Suppose d : U × U → R is a pseudodistance (definition 2.7) on an open subset U ⊆ Rn which is L-Lipschitz with respect to d0 , i.e. d ≤ Ld0 . Then there is a measurable function ρ : U × Rn → R so that for a.e. x ∈ U , i) ρ(x, ·) is a semi-norm on Rn ,


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and ii) for every v ∈ Rn − {0} we have lim sup r→0

d(y, y + rv) − ρ(x, v) | y ∈ B0 (x, r) = 0. r

(10.8)

In other words, the pseudo-distance d behaves infinitesimally like a measurable Finsler pseudo-metric. Corollary 10.9 Let U , d, and ρ : U × Rn → [0, ∞) be as in Theorem 10.7, and for every x ∈ Rn define a pseudo-metric dx,r on Rn by dx,r (y, z) = d(x + ry, x + rz). Then for a.e. x ∈ U the family of pseudo-metrics 1r dx,r converges uniformly on compact subsets of Rn × Rn as r → 0 to the pseudo-metric defined by the semi-norm ρ(x, ·). In particular, if K ⊂ Rn is a compact subset, then the pseudo-metric spaces (K, 1r dx,r , 0) converge in the Gromov-Hausdorff topology to K with the pseudo-metric determined by ρ(x, ·). Proof of Corollary 10.9. Pick x ∈ U so that (10.8) holds, and let d¯x : Rn × Rn → [0, ∞) be the pseudo-metric defined by ρ(x, ·). The pseudometrics 1r dx,r are L-Lipschitz, and by (10.8) they converge pointwise on Rn × Rn to d¯x ; therefore they converge uniformly on compact sets. If K ⊂ Rn is compact, then idK : (K, 1r dx,r ) → (K, d¯x ) is an (r)t u Hausdorff approximation where limr→0 (r) = 0. Proof of Theorem 10.7. The proof is based on a covering argument, and is analogous to the proof of the Rademacher-Stepanoff theorem on the differentiability of Lipschitz functions f : Rn → R. Since (truncated) cylinders will be used repeatedly below, we will use the notation Cyl(a, b, t), a, b ∈ Rn , t ∈ (0, ∞) to denote the cylinder with core segment ab and thickness t: Cyl(a, b, t) := {s + n | s ∈ ab, n ⊥ ab, knk0 < t}. The caps of the cylinder Cyl(a, b, t) are the subsets {a + n | n ⊥ ab, knk0 ≤ t} and {b + n | n ⊥ ab, knk0 ≤ t}. We now introduce functions that compare the pseudo-distance function d with the Euclidean distance function d0 ; these functions are analogous to directional derivatives. Define measurable functions ρ : U × Rn → R, ρ : U × Rn → R by

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d(y, y + rv) | y ∈ B0 (x, r) r

d(y, y + rv) | y ∈ B0 (x, r) . r

ρ(x, v) = lim inf r→0

and ρ(x, v) = lim sup r→0

Observe that ρ(x, ·) and ρ(x, ·) are L-Lipschitz with respect to d0 for each x ∈ U : |ρ(x, v1 ) − ρ(x, v2 )| ≤ Lkv1 − v2 k0

(10.10)

|ρ(x, v1 ) − ρ(x, v2 )| ≤ Lkv1 − v2 k0

(10.11)

and

for every v1 , v2 ∈ Rn . Given x ∈ U , v ∈ Rn − {0}, µ, ν ∈ (0, ∞), we let C(x, v, µ, ν) be the collection of cylinders Cyl(y, y + rv, rν) ⊂ U where r > 0, y ∈ B0 (x, r), and d(y, y + rv) < ρ(x, v) + µ. r

(10.12)

Note that since d ≤ Ld0 , if we take a cylinder Cyl(y, y + rv, rν) ∈ C(x, v, µ, ν), and a pair of points u, v ∈ Cyl(y, y + rv, rν) – one from each cap of Cyl(y, y + rv, rν) – then by the triangle inequality for d we have d(u, v) ≤ ρ(x, v) + µ + 2Lν. r

(10.13)

Elements of C(x, v, µ, ν) are somewhat off-centered cylinders with direction v where the d-distance between caps is approximately infimal (among such cylinders). For given v ∈ Rn − {0}, ν ∈ (0, ∞), cylinders of the form Cyl(y, y + rv, rν) with y ∈ B0 (x, r) are contained in the closed ball B0 (x, r(1 + kvk0 + ν)) and have uniform density there: (rkvk0 )(ωn−1 (rν)n−1 ) Ln (Cyl(y, y + rv, rν)) = Ln (B0 (x, r(1 + kvk0 + ν))) ωn [r(1 + kvk0 + ν)]n c(n)kvk0 ν n−1 = (10.14) (1 + kvk0 + ν)n


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Lemma 10.15 Suppose v ∈ Rn − {0} and x ∈ U is an approximate continuity point of ρ(·, v) : U → R. Then ρ(x, αv) ≤ |α|ρ(x, v) for every α ∈ R. Proof of lemma. Our goal is to show that if r ∈ (0, ∞) is sufficiently small and y ∈ B0 (x, r), then d(y,y+rαv) . |α|ρ(x, v). To prove this we thicken r the segment y(y + rαv) into the cylinder Cyl(y, y + rαv, rν), and estimate the d-distance between its caps. The lemma is obvious when v = 0 or α = 0, so we assume henceforth that v 6= 0, α 6= 0. Pick 1 ∈ (0, ∞), ν1 ∈ (0, ∞). Since x ∈ U is an approximate continuity point of ρ(·, v), the density of the set {z | |ρ(z, v) − ρ(x, v)| < 1 } in B0 (x, r) tends to 1 as r tends to zero. In view of the density estimate (10.14), the density of {z | |ρ(z, v) − ρ(x, v)| < 1 } in cylinders of form Cyl(y, y + rαv, rν1 ), y ∈ B0 (x, r), also tends to 1 as r → 0. Choose r1 ∈ (0, ∞) so that the latter density is ≥ 1 − 1 when r ≤ r1 . Fix a cylinder C = Cyl(y, y + rαv, rν1 ) where r ≤ r1 , y ∈ B0 (x, r). Let T = {z ∈ Interior(C) | |ρ(z, v) − ρ(x, v)| < 1 }. Pick µ2 , ν2 ∈ (0, ∞), and let D be the collection of cylinders in C(t, v, µ2 , ν2 ) which are contained in C, where t ranges over T . The density estimate (10.14) implies that D is a Vitali cover8 of T . So by a standard covering argument there is a disjoint subcollection D0 ⊂ D so that Ln (T \ (∪D∈D0 D)) = 0. Hence ∪D∈D0 D has density ≥ 1 − 1 in C. By Fubini’s theorem, there is a segment y 0 (y 0 + rαv) with endpoints in the caps of C so that the density of (∪D∈D0 D)∩y 0 (y 0 + rαv) in the segment y 0 (y 0 + rαv) is ≥ 1−1 . Applying the cap separation estimate (10.13), the Lipschitz estimate d ≤ Ld0 , and the triangle inequality for d, we get d(y 0 , y 0 + rαv) ≤ |α|(ρ(x, v) + 1 + µ2 + 2Lν2 ) r and so d(y, y + rαv) ≤ |α|(ρ(x, v) + 1 + µ2 + 2Lν2 ) + 2Lν1 . r

(10.16)

If ∈ (0, ∞) is given and 1 , µ2 , ν1 , ν2 are chosen so that the right hand side of (10.16) is < |α|ρ(x, v) + , and r1 is chosen accordingly, then we have d(y, y + rαv) < |α|ρ(x, v) + r provided r ≤ r1 and y ∈ B0 (x, r). This proves the lemma. t u A Vitali cover of a set S ⊂ Rn is a collection of measurable sets Yi ⊂ Rn with the following property: there is a density δ > 0 so that for every s ∈nS and every r > 0, there (Yi ) is an r0 < r and an i so that Yi ⊂ B0 (s, r0 ) and the density Ln L is > δ. (B0 (s,r 0 )) 8

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Proof of Theorem 10.7 continued. Define U0 ⊂ U to be the set of all x ∈ U which are approximate continuity points of ρ(·, v) for all v ∈ Qn ; since Qn is countable we have Ln (U − U0 ) = 0. If x ∈ U0 , then Lemma 10.15 and the triangle inequality for d imply the inequality ρ(x, v1 + v2 ) ≤ ρ(x, v1 ) + ρ(x, v2 ) for v1 , v2 ∈ Qn . If v ∈ Qn , α ∈ Q, then by Lemma 10.15 we have ρ(x, αv) ≤ ρ(x, αv) ≤ |α|ρ(x, v)

(10.17)

If |α| = 6 0 then we may replace α with α−1 in (10.17), thereby deducing that ρ(x, αv) = |α|ρ(x, v) for every α ∈ Q, v ∈ Qn . Since ρ(x, ·) : Rn → [0, ∞) is L-Lipschitz, the homogeneity and subadditivity of ρ(x, ·)|Qn extends to Rn ; so ρ(x, ·) is a semi-norm on Rn . Letting ρ = ρ, we have proved Theorem 10.7. t u Proposition 10.18 Let X be a metric space, let U ⊂ En be an open subset, and suppose f : U → X is a Lipschitz map. Then either a) There is a p ∈ f (U ), and sequences Rk ∈ (0, ∞), Sk ⊂ X so that Rk → 0, d(p, Sk ) → 0, and R1k Sk converges in the Gromov-Hausdorff topology to the unit ball in a normed space (Rn , k · k), or b) Hn (f (U )) = 0. Proof. This is a consequence of Corollary 10.9 and a covering argument. Notation. If S is a collection of subsets of a pseudo-metric space Z with distance function δ, then kSk := sup Diam(S). S∈S

If α ∈ [0, ∞), then Hδα (Z, S) :=

X

ωα [Diam(S)]α

S∈S

where ωα is a universal constant. The α-dimensional Hausdorff measure of (Z, δ) is Hδα (Z) := lim inf {Hδα (Z, S) | S is a countable cover of Z, kSk < }. →0

When δ is clear from the context we will omit it from the notation. Let dX be the distance function on X, and let d : U × U → [0, ∞) be the pullback of dX by f : d(u1 , u2 ) := dX (f (u1 ), f (u2 )).


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Applying Theorem 10.7 to d, let ρ : U × Rn → [0, ∞) be the measurable function which satisfies i) and ii) for a.e. x ∈ U , and let U0 ⊂ U be the set where i) and ii) hold. We first show that if x ∈ U0 and ρ(x, ·) : Rn → R is a norm on Rn , then a) holds. Set k · k := ρ(x, ·), and let B be the unit ball k · k−1 ([0, 1]) ⊂ Rn . By Corollary 10.9, (x + rB, 1r d) converges to B in the Gromov-Hausdorff topology, so a) holds. Let U1 be the set of x ∈ U0 for which ρ(x, ·) is not a norm. We will show that if Ln (U \ U1 ) = 0 then Hdn (U ) = 0; this clearly implies b). Lemma 10.19 Given x ∈ U1 and ∈ (0, ∞) there is an r0 ∈ (0, ∞) so that every ball B0 (x, r) with r < r0 admits a cover C with Hdn (B0 (x, r), C) < Ln (B0 (x, r))

(10.20)

Proof of lemma. Let k · k1 := ρ(x, ·), and let d1 be the pseudo-distance function on Rn associated with the semi-norm k · k1 . By assumption k · k1 is not a norm, so it is zero on some 1-dimensional subspace V ⊆ Rn . Let k·k2 : Rn /V → [0, ∞) denote the induced semi-norm on the quotient space Rn /V , and let d2 be the corresponding distance function. If λ ∈ (0, 1), the set π(B0 (0, 1)) ⊆ Rn /V can be covered by C1 ( λ1 )n−1 d2 -balls of radius λ, since (Rn /V, k · k2 ) is a normed vector space of dimension ≤ n − 1. By Corollary 10.9 the family of pseudo-metric spaces (B0 (x, r), 1r d) converges to (B0 (0, 1), d1 ), which is isometric to (π(B0 (0, 1)), d2 ) via π : Rn → Rn /V . Therefore when r is sufficiently small (B0 (x, r), d) can be covered by C1 ( λ1 )n−1 d-balls of radius 2rλ; for such a covering C we have X Hdn (B0 (x, r), C) = ωn [diamd (C)]n C∈C

≤ C2 λrn . If λ is sufficiently small (10.20) will be satisfied.

t u

Proof of Proposition 10.18 continued. Since U is a countable union of sets with finite Lebesgue measure and Hdn is countably subadditive, it suffices to treat the case that Ln (U ) < ∞. We assume that Ln (U1 ) = Ln (U ). Pick > 0. By Lemma 10.19 there is a Vitali cover D of U1 by Euclidean balls contained in U , such that each ball B ∈ D admits a cover CB satisfying a) Hdn (B, CB ) < Ln (B) and b) Diamd (C) < for every C ∈ CB . By the Vitali covering lemma there is a disjoint subcollection D0 ⊆ D so that Ln (U1 − ∪B∈D0 B) = 0. Letting C 0 = ∪B∈D0 CB we get Hdn (∪B∈D0 B, C 0 ) X Ln (B) = Ln (U ). ≤ B∈D0

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B. Kleiner

Since d ≤ Ld0 , Hdn (U − ∪B∈D0 B) ≤ (Ln )Ln (U − ∪B∈D0 B) = 0, and therefore we can find a cover C 00 of U − ∪B∈D0 B with a) Diamd C < for every C ∈ C 0 and b) Hdn (U − ∪B∈D0 B, C 00 ) < Ln (U ). So Hdn (U, C 0 ∪ t u C 00 ) ≤ 2Ln (U ), and we conclude that Hdn (U ) = 0. Proposition 10.21 Let X be an often convex space9 , let V ⊂ U ⊂ X be open subsets, and suppose Hn (U, V ) 6= {0}. Then there is a p ∈ X, and sequences Rk ∈ (0, ∞), Sk ⊂ X so that Rk → 0, d(p, Sk ) → 0, and R1k Sk converges in the Gromov-Hausdorff topology to a unit ball in a normed space (Rn , k · k). Proof. By Proposition 10.18 is suffices to show that Hn (U, V ) = {0} for every open pair in X provided the image of every Lipschitz map from Rk into X has zero k-dimensional Hausdorff measure when k ≥ n. We prove this by modifying the proofs of Lemma 5.1 and Proposition 5.2. Given a finite polyhedron P and a map σ00 of its 0-skeleton (P )0 into X, we may produce a Lipschitz map σ : P → X as follows. Let Sd P be the first barycentric subdivision of P . Extend σ00 to a map σ0 : (Sd P )0 → X by letting σ0 (v) = σ00 (w) where v ∈ (Sd P )0 \ (P )0 , and w is a vertex of the P -simplex determined by v. Inductively extend σj−1 : (Sd P )j−1 → X to σj : (Sd P )j → X by coning at barycenters. This gives us a Lipschitz map σ : P → X with the property that for each simplex τ of P , Diam(σ(τ )) ≤ Diam(σ00 (V ertex(τ ))). We may use the extension process defined above in the proof of Lemma 5.1 instead of using barycentric simplices; in particular if K ⊆ X is a compact set with zero k-dimensional Hausdorff measure, then the inclusion iK : K → X can be approximated by maps which factor as K → P → X where Dim(P ) < k. Pick α ∈ Hn (U, V ). Then α is in the image of Hn (f ) for some map of pairs f : (M, N ) → (U, V ) where M is a polyhedron of dimension n. Adapting the argument of Lemma 5.1 part 1 to our often convex space, we may approximate f with a Lipschitz map f1 ; by assumption Im f1 has zero n-dimensional Hausdorff measure, and when d(f1 , f ) is sufficiently small f1 : (M, N ) → (U, V ) will be homotopic (as a map of pairs) to f . The inclusion iIm f1 : Im f1 → X may be approximated by a map g : Im f1 → X which factors through a polyhedron of dimension < n. Hence if d(g, iIm f1 ) is sufficiently small we get Hn (f ) = Hn (f1 ) = Hn (g ◦ f ) = Hn (g) ◦ Hn (f1 ) = 0 so [α] = 0.

t u

We don’t really need X to be often convex. It’s enough to be able to cone off Lipschitz maps σ : ∆n → X at a point x ∈ X to obtain a Lipschitz σ 0 : ∆n+1 → X with Diam(Im σ 0 ) < C · Diam(Im σ). 9


449

10.3 Producing flats in convex length spaces The next result links the large-scale geometry of convex spaces with their local structure. Proposition 10.22 Let (X, d) be a locally compact convex length space with cocompact isometry group. Suppose there are sequences Rk ∈ (0, ∞), Sk ⊂ X, and a normed vector space (Rn , k · k), so that Rk → ∞, and 1 n Rk Sk converges to B(1) ⊂ (R , k · k) in the Gromov-Hausdorff topology; equivalently, suppose that (Rn , k·k) can be isometrically embedded in some asymptotic cone Xω of X. Then there is an isometric embedding of some n-dimensional Banach space in X. If n is the maximum10 dimension of Banach spaces which isometrically embed in X, then (Rn , k · k) itself can be isometrically embedded in X. Proof. The proof is very similar to the proof of Theorem 7.1: for each k we find a (approximate) minimum pk of the average distance from a finite set of points Fk ⊂ Sk , and then extract a convergent subsequence of the set of segments {pk s | s ∈ Fk } to produce a configuration of rays that “spans” a flat subspace in X. Let B ⊂ (Rn , k · k) be the unit ball k · k−1 ([0, 1]), and let dB denote the induced distance function on B. Suppose F ⊂ ∂B is a centrally symmetric (−F = F ) finite collection of points where k · k is differentiable. By assumption there are sequences Rk → ∞, k → 0 and a sequence φk : (B, Rk dB ) → (Sk , d) of Rk k -Hausdorff approximations (see definition 2.6). Define fk : X → R by fk (·) :=

1 X d(φk (x), ·). |F | x∈F

As in the proof of Theorem 7.1 (Lemma 7.3) we would like to claim that fk attains a minimum at a point close to φk (0); unfortunately the lack of uniform convexity of fk makes it difficult to control the location of the minima of fk . Instead we will work with an approximate minimum pk of fk which is close to φk (0) (Lemma 10.28). Lemma 10.23 3 fk (φk (0)) ≤ inf fk + Rk k . 2

(10.24)

If the unit ball in a Banach space V can be covered by m balls of radius 12 , then Dim(V ) ≤ m. Therefore the local compactness of X and cocompactness of Isom(X) implies that there is a bound on the dimension of Banach spaces which isometrically embed in X. 10

450

B. Kleiner

Proof of lemma. Using the fact that φk is an k Rk -Hausdorff approximation, we have d(φk (x), y) + d(y, φk (−x)) ≥ d(φk (x), φk (−x)) ≥ Rk dB (x, −x) − Rk k Rk k Rk k + Rk dB (−x, 0) − . (10.25) = Rk dB (x, 0) − 2 2 Therefore for every y ∈ X

1 X Rk k fk (y) ≥ . Rk dB (x, 0) − |F | 2

(10.26)

x∈F

And since d(φk (x), φk (0)) ≤ Rk dB (x, 0) + Rk k we have fk (φk (0)) ≤

1 X [Rk d(x, 0) + Rk k ]. |F |

(10.27)

x∈F

t u

Combining (10.26) and (10.27) we get (10.24). Pick a sequence λk ∈ (0, ∞) such that λk → ∞ and λk k → 0.

k Lemma 10.28 For each k we may choose a point pk ∈ B(φk (0), 32 R λk ) such that for every y ∈ X − {pk } we have

fk (y) − fk (pk ) ≥ −2λk k d(y, pk )

(10.29)

Proof of lemma. Consider the closed set fk (y) − fk (φk (0)) Yk := {φk (0)} ∪ y ∈ X \ {φk (0)} | ≤ −λk k . d(y, φk (0)) k By (10.24) we have Yk ⊆ B(φk (0), 32 R λk ). Therefore fk attains a minimum at some pk ∈ Yk . But then (10.29) holds for every y ∈ X, for otherwise we get

fk (y) − fk (φk (0)) = (fk (y) − fk (pk )) + (fk (pk ) − fk (φk (0))) < −λk k (d(y, pk ) + d(pk , φk (0))) ≤ −λk k (d(y, φk (0))) so y ∈ Yk and fk (y) < fk (pk ) which is absurd.

t u

After composing φk with a suitable sequence of isometries, and after passing to a suitable subsequence, we may assume that pk converges to a point p∞ and the geodesic segments pk φk (x) converge to geodesic rays


451

p∞ φ∞ (x), where φ∞ (x) ∈ ∂∞ X. Let bφ∞ (x) denote the Busemann function of the ray p∞ φ∞ (x). By Lemma 10.1 we have bφ∞ (x) ≥ lim sup[d(φk (x), ·) − d(φk (x), pk )] k→∞

for each x ∈ F . Letting 1 X [d(φk (x), ·) − d(φk (x), pk )] f¯k := fk − fk (pk ) = |F | x∈F

we have

1 X lim sup f¯k ≤ f∞ := bφ∞ (x) . |F | k→∞ x∈F

By Lemma 10.28 we have, for every y ∈ X lim inf [f¯k (y) − f¯k (pk )] k→∞

≥ lim inf (−2λk k )d(y, pk ) = 0 k→∞

so p∞ minimizes f∞ . Our next goal is to show that f∞ is minimal along each of the rays p∞ φ∞ (x), x ∈ F . Lemma 10.30 For each v ∈ ∂B let bv denote the Busemann function of the ray t 7→ tv. For every v ∈ ∂B there is a geodesic ray p∞ ξ so that for every y ∈ p∞ ξ − p∞ , and every x ∈ F we have bφ∞ (x) (y) dB (tv, x) − dB (0, x) = bx (v) ≤ lim t→0 d(p∞ , y) t

(10.31)

In other words the Busemann function bφ∞ (x) decreases at least as fast along p∞ φ∞ (x) as dB (x, ·) (initially) decreases along 0v (this rate is the same as the value of bx at v). Proof of lemma. Pick α, l ∈ (0, ∞). We will first show that there is a yl ∈ X so that d(p∞ , yl ) = l and bφ∞ (x) (yl ) d ≤ dB (tv, 0)|t=0 + α d(p∞ , yl ) dt

for every x ∈ F

d B (0,x) Choose t > 0 small enough that dB (tv,x)−d < dt dB (tv, x) + α for all t x ∈ F . Since φk : (B, Rk dB ) → (Sk , d) is an k Rk -Hausdorff approximation, d(φk (tv), φk (x)) − d(φk (0), φk (x)) lim k→∞ Rk

452

B. Kleiner

The local structure of length spaces with curvature bounded above

The local structure of length spaces with curvature bounded above

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