Abstract
A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called Morse flats. We show that the Tits boundary ∂TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂TF. More precisely, we show that the set of singular points in ∂TF can be covered by finitely many round spheres of positive codimension.
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1 Introduction
1.1 Main results
A Hadamard manifold is a complete simply connected Riemannian manifold of non-positive sectional curvature. It has higher rank if every complete geodesic line is contained in an isometrically embedded Euclidean space of dimension at least two, a so-called flat. The structure of manifolds of non-positive curvature and higher rank was clarified in the eighties, in the series of papers [BBE85, BBS85], culminating in Ballmann’s celebrated Higher Rank Rigidity Theorem [Bal85, BS87, EH90]: If a higher rank Hadamard manifold admits quotients of finite volume, then it is either a Riemannian symmetric space or splits as a metric product. So in the presence of enough symmetry, higher rank implies special geometry. In particular, the theorem identifies higher rank as the exceptional case.
The modern theory of non-positive curvature focuses on metric spaces instead of manifolds, where the curvature condition is expressed by triangle comparison. The definition is due to Alexandrov, already in the early fifties, and later gained considerable momentum from Gromov’s seminal paper [Gro87]. Nowadays, CAT(0) spaces – the synthetic versions of Hadamard manifolds – play an important role in mathematics beyond geometry and topology. They appear in many different fields such as geometric group theory, representation theory, arithmetic and optimization. For an account of the huge amount of literature on metric spaces with synthetic curvature bounds, see the bibliographies in [AKP19, Bal04, BH99]. The following conjecture, formulated by Ballmann and Buyalo in [BB08], predicts a generalization of the Higher Rank Rigidity Theorem from Hadamard manifolds to CAT(0) spaces. It is the main motivation for the present paper.
Conjecture 1
Higher Rank Rigidity
Let X be a locally compact geodesically complete CAT(0) space. Suppose that Γ is a group of isometries of X with limit set Λ(Γ)=∂∞X. Then \(\operatorname{diam}(\partial_{T}X)=\pi \) implies that X is a Riemannian symmetric space or a Euclidean building of rank at least 2, or that X non-trivially splits as a metric product.
The condition on the diameter expresses higher rank, as it is equivalent to saying that any complete geodesic bounds a flat half-plane, cf. Sect. 2.4. If a group Γ acts on a CAT(0) space X, then its limit set Λ(Γ) is the set of accumulation points of a Γ-orbit in the ideal boundary ∂∞X. The main example of group actions on CAT(0) spaces with full limit set are geometric actions, meaning cocompact and properly discontinuous actions by isometries.
For singular spaces, Higher Rank Rigidity is wide open. It has only been established in the following short list of special cases:
-
for a 2-dimensional simplicial complex with a piecewise smooth CAT(0) metric and a geometric group action [BB96];
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for a 3-dimensional simplicial complex with a piecewise Euclidean CAT(0) metric and a geometric group action [BB00];
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for a finite dimensional CAT(0) cube complex with a geometric group action [CS11];Footnote 1
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for a locally compact and geodesically complete CAT(0) space such that its full isometry group does not fix a point at infinity and which supports a cocompact isometric action by an amenable locally compact group [CM15];
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for a locally compact and geodesically complete CAT(0) space with 1-dimensional Tits boundary which supports a geometric group action Γ↷X such that the induced action on the ideal boundary is not minimal, i.e. ∂∞X contains a proper closed Γ-invariant subset [Ric19].
The paper [Ric19] in the last item relies heavily on [GS13] where the authors developed a general strategy to approach Ballmann’s conjecture based on analyzing the dynamics of the induced action at infinity Γ↷∂∞X.
Note also that the interesting case in the next to last item concerns non-discrete groups since the only locally compact geodesically complete CAT(0) space which admits a cocompact isometric action by an amenable discrete group is the flat Euclidean space [AB98, Corollary C].
Theorem A
Let X be a locally compact CAT(0) space whose Tits boundary has dimension n−1≥1. Suppose that every geodesic in X lies in an n-flat. If X contains a periodic n-flat, then X is a Riemannian symmetric space or a Euclidean building, or X splits as a metric product.
If the isometry group of X acts cocompactly, then the condition on the dimension of the Tits boundary simply means that X does not contain an (n+1)-flat [Kle99, Theorem C]. Thus, we obtain:
Corollary B
Let X be a locally compact CAT(0) space whose isometry group acts cocompactly. Suppose that every geodesic in X lies in an n-flat but X does not admit (n+1)-flats. If X contains a periodic n-flat, then X is a Riemannian symmetric space or a Euclidean building, or X splits as a metric product.
Another consequence of Theorem A is the following.
Corollary C
Let X be a locally compact CAT(0) space whose Tits boundary has dimension n−1≥1. Suppose that every geodesic in X lies in an n-flat. If X contains a periodic n-flat, then X has a unique decomposition into irreducible factors
where Xa are rank 1 spaces, Yb are irreducible Euclidean buildings of higher rank and Mc are irreducible Riemannian symmetric spaces of higher rank.
Note that the assumptions above imply that the factors Xa are not only rank 1 but have discrete Tits boundaries. Thus if X admits a cocompact group action, then every Xa is Gromov-hyperbolic.
Recall that in the smooth case, the higher rank property can be expressed by asking that every complete geodesic admits n linearly independent parallel Jacobi fields for some n≥2. In the presence of a geometric group action these Jacobi fields can then be “integrated” to n-flats [Bal95, Section IV.4]. Hence, in the context of metric spaces, the assumption on the existence of n-flats in Theorem A is natural.Footnote 2 Recall that a flat is periodic, if its stabilizer in the isometry group of the surrounding space contains a subgroup acting geometrically on it. By [BH99, Flat Torus Theorem 7.1], the existence of a periodic n-flat in X is equivalent to the presence of a subgroup \(\mathbb{Z}^{n}<\mathop{\hbox{Isom}}(X)\). We point out that Theorem A does not assume any symmetries besides the existence of a single periodic n-flat. Also, if X is a rank n symmetric space or Euclidean building, and Γ acts cocompactly on X, then X contains a periodic n-flat [BB96, Theorem 8.9], [Mos73, Lemma 8.3], [PR72, Theorem 2.8]. We emphasize that some symmetry is required in order for the conclusion of Higher Rank Rigidity to hold true, since a geodesically complete CAT(0) space whose Tits boundary ∂TX has diameter π does not have to be a product or a Euclidean building [BB00].
We now discribe the main novelty of the present paper. Recall that a point in a metric space is called regular, if it has a neighborhood homeomorphic to an open set in a Euclidean space. The crucial ingredient behind Theorem A is a contribution to the following general conjecture which has been completely open.
Conjecture 2
Lytchak’s Regular Point Conjecture
Let X be a locally compact CAT(0) space with a geometric group action. Then ∂TX contains a regular point.
Gromov first raised the question which CAT(1) spaces do contain regular points [Gro93, p.90]. A general CAT(1) space does not have to contain any regular points. On the other hand, locally compact and geodesically complete CAT(1) spaces always do, as shown by Lytchak-Nagano in [LN19], following ideas of Burago-Gromov-Perelman [BGP92]. However, the Tits boundary of a locally compact CAT(0) space with a geometric group action is typically not locally compact.
Recall that a flat in a CAT(0) space is called Morse if it does not bound a flat half-space. We prove:
Theorem D
[Theorem 4.10] Let X be a locally compact CAT(0) space. Suppose that X contains a periodic Morse flat F. Then the complement in ∂TF of the set of regular points is a finite union of round spheres of positive codimension.
In particular, the singular set in the Tits boundary of a periodic Morse (n+1)-flat is given by a union of finitely many hyperspheres and a closed set of finite (n−2)-dimensional Hausdorff measure. In this sense the Tits boundary of a periodic Morse flat resembles a Coxeter complex. This will be used in [Sta221] in order to control the geometry of certain submetries of the Tits boundary. Let us emphasize again that Theorem D does not assume any symmetry besides the periodic Morse flat, no geometric group action is required.
The present article is the first in a series motivated by the Higher Rank Rigidity Conjecture. The sequels all rely on the crucial new idea on how to find regular points at infinity presented here. In [Sta221] we prove the following rigidity result which generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds.
Theorem 1.1
[Sta221, Main Theorem]
Let X be a locally compact CAT(0) space with a geometric group action Γ↷X. Suppose that there exists n≥2 such that every geodesic in X lies in an n-flat. If X contains a periodic Morse n-flat, then X is a Riemannian symmetric space or a Euclidean building or X non-trivially splits as a metric product.
In [Sta222] we confirm Higher Rank Rigidity as well as the Closing Lemma for spaces without 3-flats. More precisely, we obtain
Theorem 1.2
[Sta222, Theorems A and B]
Let X be a locally compact geodesically complete CAT(0) space without 3-flats. Suppose X admits a geometric group action Γ↷X. If X contains a complete geodesic which does not bound a flat half-plane, then it also contains a Γ-periodic geodesic which does not bound a flat half-plane. If every complete geodesic in X bounds a flat half-plane, then X is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product.
2 Preliminaries
2.1 Metric spaces
We assume that the reader is familiar with the geometry of metric spaces with upper curvature bound, as references we mention [AKP19, Bal04, BH99, KL97]. For the required preliminaries on buildings we refer the reader to [KL97, Lee00].
Here we only want to agree on notation and collect some basic facts that will be used later.
Euclidean n-space with its flat metric will be denoted by \(\mathbb{R}^{n}\). The unit sphere \(S^{n-1}\subset \mathbb{R}^{n}\) equipped with the induced metric will be referred to as a round sphere. Its intersection with a half-space \(\mathbb{R}^{n-1}\times [0,\infty )\) is called a round hemisphere. We denote the distance between two points x and y in a metric space X by |x,y|. For x∈X and r>0, we denote by Br(x) and \(\bar{B}_{r}(x)\) the open and closed r-ball around x, respectively. Similarly, Nr(A) and \(\bar{N}_{r}(A)\) denote the open and closed r-neighborhood of a subset A⊂X, respectively. A geodesic is an isometric embedding of an interval. It is called a geodesic segment, if it is compact. The endpoints of a geodesic segment c are denoted by ∂c.
A branch point is a point through which a certain geodesic can be extended in more than one way. More precisely, a point b is a branch point if there are geodesic segments c, c+ and c− such that c∪c± is a geodesic and c+∩c−={b}. In a geodesic metric space any pair of points is connected by a geodesic. Such a space is geodesically complete if every geodesic segment is contained in a complete local geodesic.
2.2 Spaces with an upper curvature bound
For any CAT(κ) space X, the angle between each pair of geodesics starting at the same point is well defined. Let x, y, z be three points at pairwise distance less than \(\frac{\pi}{\sqrt{\kappa}}\) in a CAT(κ) space X. Whenever x≠y, the geodesic between x and y is unique and will be denoted by xy. For y,z≠x, the angle at x between xy and xz will be denoted by ∠x(y,z). The space of directions or link at a point x∈X will be denoted by (ΣxX,∠), its elements are called directions (at x).
The dimension dim(X) of a CAT(κ) space X refers to Kleiner’s geometric dimension [Kle99]. It vanishes precisely when the space is discrete. In general, it is defined inductively:
2.3 CAT(1) spaces
For two CAT(1) spaces Z1 and Z2 we denote by Z1∘Z2 their spherical join. It is a CAT(1) space of diameter π. A pair of points ξ−, ξ+ in a CAT(1) space Z are called antipodes, if their distance is at least π.
A subset A⊂Z is called spherical, if it embeds isometrically into a round sphere. A (spherical) n-lune of angle θ∈[0,π] in a CAT(1) space is a closed convex subset λ isometric to Sn−1∘[0,θ]. Bigons in CAT(1) spaces lead to spherical lunes [BB99, Lemma 2.5]. We will use the following more general result which follows immediately from [BB99, Lemma 2.5] and [Lyt05, Lemma 4.1].
Lemma 2.1
Lune Lemma
Let ξ and \(\hat{\xi}\) be antipodes at distance π in a CAT(1) space Z. If γ± are geodesics from ξ to \(\hat{\xi}\), then γ−∪γ+ is a spherical subset in Z. More precisely, if ∠ξ(γ−,γ+)≥π, then γ−∪γ+ is a round 1-sphere in Z; and if ∠ξ(γ−,γ+)=θ<π, then γ−∪γ+ bounds a spherical lune of angle θ. Similarly, let τ±⊂Z be round n-hemispheres with ∂τ−=∂τ+ and let ζ±∈Z denote their centers. If |ζ−,ζ+|≥π, then τ−∪τ+ is a round n-sphere in Z; and if |ζ−,ζ+|=θ<π, then τ−∪τ+ bounds a spherical (n+1)-lune of angle θ.
Definition 2.2
Let Z be a CAT(1) space. We call a point ξ∈Z k-regular, if it has a neighborhood homeomorphic to an open set in \(\mathbb{R}^{k}\). The point ξ is called k-spherical, if there exists a radius r>0 such that Br(ξ)⊂Z is isometric to an r-ball in Sk. In both definitions, if it’s clear from the context, we neglect the quantification and speak of regular respectively spherical points.
Lemma 2.3
[BL06, Lemma 2.1]
Let Z be a CAT(1) space of dimension n. If σ⊂Z is a round n-sphere, then every point ξ∈Z has an antipode in σ.
This implies the following well-known criterion for geodesic completeness.
Lemma 2.4
Let Z be a CAT(1) space of dimension n. If every point ξ∈Z is contained in a round n-sphere in Z, then Z is geodesically complete.
Proof
If ξ is a point in Z contained in a round n-sphere σ⊂Z, then any local geodesic ending in the point ξ can be extended beyond ξ by a geodesic in σ, since any direction in ΣξZ has an antipode in Σξσ≅Sn−1 by Lemma 2.3. □
A simple but useful property in CAT(1) spaces with a geodesically complete link is the following.
Lemma 2.5
Let Z be a CAT(1) space and z∈Z a point where the link ΣzZ is geodesically complete. Then for every pair of directions v,w∈ΣzZ there exists an antipode \(\hat{w}\in \Sigma _{z} Z\) of w, such that
Proof
If ∠z(v,w)=π, then we can take \(\hat{w}=v\). Otherwise, we extend the geodesic from w to v up to distance π. The endpoint \(\hat{w}\) is then an antipode of w with the required property. □
The proof of Theorem A relies on the following rigidity result of Lytchak.
Theorem 2.6
[Lyt05, Main Theorem]
Let Z be a finite dimensional geodesically complete CAT(1) space. If Z has a proper closed subset A containing with each point all of its antipodes, then Z is a spherical join or a spherical building.
2.4 CAT(0) spaces
The ideal boundary of a CAT(0) space X, equipped with the cone topology, is denoted by ∂∞X. If X is locally compact, then ∂∞X is compact. The Tits boundary of X is denoted by ∂TX, it is the ideal boundary equipped with the Tits metric |⋅,⋅|T. Recall that the Tits metric is the intrinsic metric associated to the Tits angle.
If C is a closed convex subset, then it is CAT(0) with respect to the induced metric. In this case, C admits a 1-Lipschitz retraction πC:X→C. If C1 and C2 are closed convex subsets, then the distance function \(d(\cdot ,C_{1})|_{C_{2}}\) is convex; and constant if and only if \(\pi _{C_{1}}\) restricts to an isometric embedding on C2. We call C1 and C2 parallel, C1∥C2, if and only if \(d(\cdot ,C_{1})|_{C_{2}}\) and \(d(\cdot ,C_{2})|_{C_{1}}\) are constant. Let Y⊂X be a geodesically complete closed convex subset. Then we define the parallel set P(Y) as the union of all closed convex subsets parallel to Y. The parallel set is closed, convex and splits canonically as a metric product
where the cross section CS(Y) is a closed convex subset [KL97, Sect. 2.3.3].
If X1 and X2 are CAT(0) spaces, then their metric product X1×X2 is again a CAT(0) space. We have ∂T(X1×X2)=∂TX1∘∂TX2 and \(\Sigma _{(x_{1},x_{2})}(X_{1}\times X_{2})=\Sigma _{x_{1}} X_{1}\circ \Sigma _{x_{2}} X_{2}\). If X is a geodesically complete CAT(0) space, then any join decomposition of ∂TX is induced by a metric product decomposition of X [KL97, Proposition 2.3.7].
The Tits boundary of a closed convex subset Y⊂X embeds canonically ∂TY⊂∂TX. If two closed convex subsets Y1 and Y2 intersect in X, then
For points ξ∈∂∞X and x∈X we denote the Busemann function centered at ξ based at x by bξ,x. If ρ:[0,∞)→X denotes the geodesic ray asymptotic to ξ and with ρ(0)=x, then
It is a 1-Lipschitz convex function whose negative gradient at a point y∈X is given by logy(ξ). We denote the horoball centered at a point ξ∈∂∞X and based at the point x∈X by
It is a closed convex subset with
A n-flat F in a CAT(0) space X is a closed convex subset isometric to \(\mathbb{R}^{n}\). In particular, ∂TF⊂∂TX is a round (n−1)-sphere. On the other hand, if X is locally compact and σ⊂∂TX is a round (n−1)-sphere, then either there exists an n-flat F⊂X with ∂TF=σ, or there exists a round n-hemisphere τ+⊂∂TX with σ=∂τ+ [Lee00, Proposition 2.1]. Consequently, if ∂TX is (n−1)-dimensional, then any round (n−1)-sphere in ∂TX is the Tits boundary of some n-flat in X. Moreover, a round n-hemisphere τ+⊂∂TX bounds a flat (n+1)-half-space in X if and only if its boundary ∂τ+ bounds an n-flat in X. A flat is called Morse, if it does not bound a flat half-space.Footnote 3
The following basic fact will be used repeatedly throughout the paper.
Lemma 2.7
[Lee00, Sublemma 2.3]
Let F be a flat in a CAT(0) space and let C⊂X be a closed convex subset so that ∂∞F⊂∂∞C. Then C contains a flat F′ parallel to F.
A flat (n-dimensional) half-space H⊂X is a closed convex subset isometric to a Euclidean half-space \(\mathbb{R}_{+}^{n}\). Its boundary ∂H⊂X is an (n−1)-flat and its Tits boundary is a round (n−1)-hemisphere ∂TH⊂∂TX. Flat half-spaces will play a central role in our arguments later, and we agree to denote them by H and their boundaries by ∂H=h.
We define the parallel set of a round sphere σ⊂∂TX as
where F is a flat in X with ∂TF=σ, if such a flat exists.
The proof of Theorem A relies on the following rigidity result of Leeb.
Theorem 2.8
[Lee00, Main Theorem]
Let X be a locally compact, geodesically complete CAT(0) space. If ∂TX is a connected thick irreducible spherical building, then X is either a symmetric space or a Euclidean building.
3 Recognizing a building at infinity
The aim of this section is to prove the following result.
Theorem 3.1
Let X be a locally compact CAT(0) space where every geodesic lies in an n-flat. Suppose that dim(∂TX)=n−1. If ∂TX contains an open relatively compact subset U, then ∂TX is a spherical building.
Note that the assumptions imply that X is geodesically complete since we can extend every geodesic segment inside a suitable n-flat to a complete geodesic line.
The result is similar to [BL06, Theorem 1.6] and builds on Theorem 2.6 in the same way. Note that if we would know that any pair of antipodes in ∂TX lies in a round (n−1)-sphere, then we could directly apply [BL06, Theorem 1.6].
Using the structure theory of locally compact and geodesically complete spaces with upper curvature bounds [LN19], one can deduce Theorem 3.1 from the following similar result which will suffice for our proof of Theorem A.
Theorem 3.2
Let X be a locally compact CAT(0) space where every geodesic lies in an n-flat. Suppose that dim(∂TX)=n−1. If ∂TX contains an (n−1)-spherical point, then ∂TX is a spherical building.
We need some preparation before we can provide the proof of Theorem 3.1.
Recall that a pair of antipodes in the Tits boundary of a CAT(0) space X does not have to bound a complete geodesic in X. Nevertheless, for locally compact geodesically complete spaces we have the following.
Lemma 3.3
Let X be a locally compact, geodesically complete CAT(0) space. Let ξ and \(\hat{\xi}\) be antipodes in ∂TX. Then there exists a sequence of complete geodesics ck in X with \(\partial_{\infty}c_{k}=\{\hat{\xi},\xi _{k}\}\) and ξk→ξ in ∂∞X.
Proof
If \(|\xi ,\hat{\xi}|>\pi \), then there is a complete geodesic c in X with \(\partial_{\infty}c=\{\xi ,\hat{\xi}\}\) [Lee00, Proposition 2.1]. Hence, we may assume \(|\xi ,\hat{\xi}|=\pi \). Fix a base point o∈X and let ρ:[0,∞)→X be a parametrization of the geodesic ray oξ. Since X is locally compact and geodesically complete, all links in X are geodesically complete [LN19, Corollary 5.8, Corollary 5.9]. By Lemma 2.5, we can choose for each \(k\in \mathbb{N}\) a complete geodesic ck extending the geodesic ray \(\rho (k)\hat{\xi}\) and such that
where ξk:=ck(+∞). Now for every p∈oξ we have
Since \(\angle _{\rho (k)}(\xi ,\hat{\xi})\to |\xi ,\hat{\xi}|=\pi \). As p∈oξ was arbitrary, this shows ξk→ξ in ∂∞X. Indeed, by local compactness, we may assume ξk→ξ′ after passing to a subsequence. In particular, ∠x(ξk,ξ′)→0 for all x∈X. Now if ξ≠ξ′, then there exists q∈oξ such that ∠q(ξ,ξ′)>0. But
□
Lemma 3.4
Let X be a locally compact CAT(0) space. Suppose that every geodesic in X lies in an n-flat. If ξ∈∂TX is an (n−1)-spherical point, then for every antipode \(\hat{\xi}\) of ξ there exists a round (n−1)-sphere σ⊂∂TX containing ξ and \(\hat{\xi}\).
Proof
Let ξ∈∂TX be an (n−1)-spherical point, cf. Definition 2.2. Then there exists s>0 such that Bs(ξ) is contained in any (n−1)-sphere which contains ξ. If \(\hat{\xi}\in \partial_{T}X\) is an antipode of ξ, then by Lemma 3.3, there exists a sequence of complete geodesics (ck) in X with \(\partial_{\infty}c_{k}=\{\xi ,\hat{\xi}_{k}\}\) and \(\hat{\xi}_{k}\to \hat{\xi}\). By assumption, there exists n-flats Fk with ck⊂Fk. Hence, if η∈∂TX is a point with |η,ξ|=s, then \(|\eta ,\hat{\xi}_{k}|=\pi -s\) for all \(k\in \mathbb{N}\). Thus,
by lower semi-continuity of the Tits metric. We obtain \(|\eta ,\hat{\xi}|=\pi -s\) for every point η at distance s from ξ. In particular, through every such point runs a geodesic from ξ to \(\hat{\xi}\). Denote by σ⊂∂TX the union of all these geodesics. It follows from Lemma 2.1 that σ is a round (n−1)-sphere. □
Proposition 3.5
Let X be a locally compact CAT(0) space where every geodesic lies in an n-flat. Suppose dim(∂TX)=n−1. Then the set O⊂∂TX of (n−1)-spherical points is closed under taking antipodes.
Proof
Let ξ be a (n−1)-spherical point in ∂TX and let \(\hat{\xi}\) be an antipode. By Lemma 3.4, there exists a round (n−1)-sphere σ⊂∂TX which contains ξ and \(\hat{\xi}\). Since ξ is spherical, there exists s>0 such that Bs(ξ)⊂σ. We claim that \(B_{s}(\hat{\xi})\subset \sigma \) holds as well. Let \(\hat{\eta}\) be a point in \(B_{s}(\hat{\xi})\). Since dim(∂TX)=n−1, we can extend the geodesic \(\hat{\eta}\hat{\xi}\), as in the proof of Lemma 2.4, up to an antipode η of \(\hat{\eta}\) in σ. Then \(|\eta ,\xi |=|\hat{\eta},\hat{\xi}|< s\) and therefore η is spherical. By Lemma 3.4, we find a round (n−1)-sphere σ′⊂∂TX which contains η and \(\hat{\eta}\). Because η∈Bs(ξ), we must have Bs(ξ)⊂σ∩σ′. By construction, \(\hat{\xi}\) also lies in σ∩σ′. Now convexity of σ∩σ′ implies σ=σ′ and therefore \(\hat{\eta}\in \sigma \) as required. □
Lemma 3.6
Let X be a locally compact CAT(0) space with dim(∂TX)=n−1 and where every geodesic lies in an n-flat. Suppose X splits as a non-trivial metric product X≅X1×X2. Then we have n=n1+n2 where dim(∂TXj)=nj−1, j=1,2, and every complete geodesic in Xj is contained in an nj-flat.
Proof
It is clear that the rank is additive. So let cj be a complete geodesic in Xj and let \(\tilde{c}=(c_{1},c_{2})\) be the diagonal geodesic in c1×c2⊂X1×X2. If \(\tilde{F}\) is an n-flat in X which contains \(\tilde{c}\), then by [KL97, Lemma 2.3.8] there are flats Fj⊂Xj such that \(\tilde{F}\subset F_{1}\times F_{2}\). In particular, dim(Fj)=nj and cj⊂Fj. □
Proof of Theorem 3.2
By assumption, dim(∂TX)=n−1 and every point ξ in ∂TX lies in a round (n−1)-sphere. Hence, Lemma 2.4 implies that ∂TX is geodesically complete.
We proceed by induction on the dimension of ∂TX. If the dim(∂TX) is 0, then ∂TX is discrete and there is nothing to show. So let’s assume dim(∂TX)≥1. Recall that two antipodes ξ, \(\hat{\xi}\) in ∂TX at distance >π bound a complete geodesic in X [Lee00, Proposition 2.1]. Since every complete geodesic in X lies in an n-flat, we conclude that the diameter of ∂TX is π.
Suppose X splits as a non-trivial metric product X≅X1×X2. Then by Lemma 3.6, the factors are subject to the induction hypothesis and we may assume that X is irreducible.
By assumption, the set of (n−1)-spherical points O⊂∂TX is non-empty. If O=∂TX, then X is isometric to \(\mathbb{R}^{n}\). So let us assume that O is a proper subset of ∂TX. By Proposition 3.5, the set ∂TX∖O is a non-empty, closed proper subset which contains with every point also all its Tits antipodes. Since ∂TX is irreducible, Theorem 2.6 implies that it is a spherical building. □
Proof of Theorem 3.1
By Theorem 3.2, it is enough to find an (n−1)-spherical point in ∂TX. If U is an open relatively compact subset of ∂TX, then U contains an open subset U′ homeomorphic to an n-manifold [LN19, Theorem 1.2]. Hence, for a point ξ∈U′ there exists an ϵ>0 such that Bϵ(ξ)⊂U′. If σ⊂∂TX denotes a round (n−1)-sphere with ξ∈σ, then σ∩Bϵ(ξ)=Bϵ(ξ). Hence ξ is (n−1)-spherical. □
4 Rank n spaces with periodic n-flats
4.1 From branch points to orthogonal half-planes
Definition 4.1
Let F⊂X be a flat in a CAT(0) space. Then we call a flat half-space H⊂X a flat half-space orthogonal to F, or, if the respective flat is clear from the context, an orthogonal flat half-space if
where the angle is measured in the cross section of the parallel set P(∂H). In case H is 2-dimensional, we adjust to flat half-plane orthogonal to F and orthogonal flat half-plane, respectively.
Definition 4.2
Let X be a locally compact CAT(0) space and F⊂X a flat. Let Isom(X) denote the isometry group of X. We say that F is periodic, if its stabilizer Stab(F):={γ∈Isom(X)| γ(F)=F} contains a subgroup acting geometrically on F.
Lemma 4.3
Let X be a locally compact CAT(0) space and let F⊂X be a periodic n-flat with n≥2. Suppose that F=F−×F+ is a metric product decomposition into flats F± of dimension n±. Then F± is a periodic n±-flat in the cross section CS(F∓).
Proof
By Bieberbach’s theorem, the stabilizer of F contains a free abelian subgroup A of rank n which acts cocompactly on F by Euclidean translations. Since A preserves ∂TF pointwise, it also preserves P(F∓) and the splitting P(F∓)≅F∓×CS(F∓). Hence its projection to CS(F∓) contains a free abelian subgroup of rank n±=n−n∓ which acts geometrically by Euclidean translations on F±⊂CS(F∓). □
The following result implies that a branch point in the ideal boundary of a periodic flat leads to an orthogonal flat half-plane.
Lemma 4.4
[HK05, Lemma 2.3.1]
Let X be a locally compact CAT(0) space. Let F⊂X be a periodic flat. Let ξ∈∂TF be a point such that there exists a sequence (ξk) in ∂TX∖∂TF which converges to ξ with respect to the Tits metric. Then there exists a flat half-plane H⊂X asymptotic to ξ and orthogonal to F (cf. Definition 4.1).
Proof
Since F is periodic, it is enough to find for every R>0 a flat strip \(S\cong \mathbb{R}\times [0,R]\) which is asymptotic to ξ and orthogonal to F. We fix a base point o∈F. For every \(k\in \mathbb{N}\) we choose xk∈oξk such that |xk,F|=R. Denote by \(\bar{x}_{k}\in F\) the nearest point to xk. Choose isometries γk∈Stab(F) such that all points \(\gamma _{k}(\bar{x}_{k})\) lie in a fixed compact set. After passing to a subsequence, we obtain a limit geodesic c∞ which is parallel to F and spans the required flats strip. More precisely, we have the convergence γk(oξk,xk)→(c∞,x∞) with respect to pointed Hausdorff topology. The complete geodesic c∞ is asymptotic to ξ since |γkξk,ξ|=|ξk,ξ|→0. Because \(x_{k} o\subset \bar{N}_{R}(F)\), also c∞ is contained in the R-neighborhood of F and has even distance R from F since |x∞,F|=R. Denote by \(\bar{c}_{\infty}\subset F\) the complete geodesic which contains the point x∞ and is asymptotic to ξ. Then c∞ and \(\bar{c}_{\infty}\) span the required flat strip by [KL97, Lemma 2.3.5]. □
Corollary 4.5
Let X be a locally compact CAT(0) space. Let \(\hat{F}\subset X\) be a periodic (n+1)-flat with \(\partial_{T}\hat{F}=\hat{\sigma }\). Denote by λ⊂∂TX a spherical (k+1)-lune with k≤n bounded by round k-hemispheres τ±. Suppose that \(\lambda \cap \hat{\sigma }=\tau ^{-}\). Then there exists a round (k+1)-hemisphere \(\hat{\tau}\subset \partial_{T}X\) such that \(\partial \hat{\tau}\subset \hat{\sigma }\) and \(\tau ^{-}\subset \partial \hat{\tau}\). Moreover, \(\hat{\tau}\) is the ideal boundary of a flat half-space \(\hat{H}\) orthogonal to \(\hat{F}\).
Proof
Set σ=∂τ± and write \(\hat{\sigma }=\sigma \circ \sigma ^{\perp}\). Let \(\hat{F}=F\times F^{\perp}\) be a corresponding product splitting. We have λ=σ∘[0,θ]. By Lemma 4.3, F⊥ is a periodic flat in the cross section CS(F). Denote by ζ− the center of τ−. Then the cross section of λ provides a path in ∂TCS(F) which intersects ∂TF⊥ only in ζ−. Thus, by Lemma 4.4, there exists an orthogonal flat half-plane H⊂CS(F) for F⊥ which is asymptotic to ζ−. Hence, \(\hat{H}:=F\times H\subset P(F)\) and \(\hat{\tau}:=\partial_{T}\hat{H}\) define an orthogonal flat half-space and round (k+1)-hemisphere, respectively, as required. □
4.2 Orthogonal half-spaces from a sequence of lower dimensional ones
In this section we prove the following key result which allows us to understand the singular set in the Tits boundary of a periodic Morse flat. The responsible geometric property behind the proof is flatness of intersecting parallel sets, similar as in [Sta15, Theorem 2].
Proposition 4.6
Let X be a locally compact CAT(0) space. Let \(\hat{F}\) be a periodic \(\hat{n}\)-flat in X with \(\partial_{T}\hat{F}=\hat {\sigma }\). Suppose that there is a sequence of flat half-spaces (Hk) of dimension d≤n, all orthogonal to \(\hat{F}\), such that their boundary flats hk:=∂Hk are pairwise non-parallel. Then there exists a (d+1)-dimensional flat half-space \(\hat{H}\) orthogonal to \(\hat{F}\). Moreover, if the sequence of round spheres σk=∂∞hk converges to a round sphere \(\sigma \subset \hat{\sigma }\) and for every \(k\in \mathbb{N}\) the convex hull of σk and σ is a round sphere of dimension \(\hat{d}-1\), then we can arrange \(\hat{H}\) to have dimension \(\hat{d}+1\) and such that \(\sigma \subset \partial_{\infty}\hat{H}\).
Proof
Since \(\hat{F}\) is periodic, we may assume that we have a convergent sequence of pairwise distinct pointed flat half-spaces (Hk,pk)→(H∞,p∞), where pk∈Fk. The limit H∞ with boundary flat h∞:=∂H∞ is then still orthogonal to \(\hat{F}\). Set \(\partial_{T}H_{k}=\tau _{k}^{+}\), \(\partial_{T}H_{\infty}=\tau _{\infty}^{+}\) and denote by ζk and ζ∞ the respective centers. Because p∞∈h∞ and H∞ is orthogonal to \(\hat{F}\), the Busemann function \(b_{\zeta _{\infty},p_{\infty}}\) is non-negative on \(\hat{F}\). Since the geodesic rays pkζk converge to p∞ζ∞, we find a sequence tk→∞ and points xk∈pkζk such that \(b_{\zeta _{\infty},p_{\infty}}(x_{k})\leq -t_{k}\). The geodesic ray pkζk lies in Hk and therefore xk lies in P(hk). Thus HB(ζ∞,xk)∩P(hk) is a non-empty closed convex set and we have
As \(\hat{F}\subset P(h_{k})\), we see ∂∞h∞⊂∂∞(HB(ζ∞,xk)∩P(hk)). From Lemma 2.7 we infer that there is a (d−1)-flat Ek⊂HB(ζ∞,xk)∩P(hk) with ∂∞Ek=∂∞h∞. Since the flats hk are pairwise non-parallel and Ek⊂P(hk), we see that there is a flat Fk in P(hk) of dimension at least d which contains Ek and such that ∂∞hk⊂∂∞Fk. Note that \(b_{\zeta _{\infty},p_{\infty}}\) is bounded above on Ek by −tk since Ek⊂HB(ζ∞,xk). As \(b_{\zeta _{\infty},p_{\infty}}\) is non-negative on \(\hat{F}\) we conclude \(E_{k}\cap N_{t_{k}}(\hat{F})=\emptyset \). Hence if \(F'_{k}\subset \hat{F}\) denotes the closest parallel flat to Fk, then \(|F_{k},F'_{k}|_{H}\geq t_{k}\). In particular, Fk and \(F'_{k}\) bound a flat strip of dimension ≥d+1, orthogonal to \(\hat{F}\) and of width at least tk. Since \(\hat{F}\) is periodic and tk→∞, we obtain a flat half-space \(\hat{H}\) orthogonal to \(\hat{F}\) and of dimension at least d+1 as a limit.
To prove the finer statement, simply note that in the argument above the flat Ek⊂P(hk) spans a flat Fk⊂P(hk) of dimension \(\hat{d}\) if and only if the convex hull of the spheres ∂∞Ek=∂∞h∞ and ∂∞hk is a round sphere of dimension \(\hat{d}-1\) in \(\hat{\sigma }\). The rest of the argument stays the same. □
4.3 Tits boundary of a periodic Morse flat
In this final section we provide proofs of our main results.
Definition 4.7
Let X be a locally compact CAT(0) space. Let \(\hat{F}\subset X\) be a periodic flat with \(\partial_{T}\hat{F}=\hat{\sigma }\). A round sphere \(\sigma \subset \hat{\sigma }\) is called singular, if there exists a round hemisphere τ+⊂∂TX with ∂τ+=σ and \(\tau ^{+}\cap \hat{\sigma }=\sigma \) such that the union τ+∪τ− is a round sphere for every round hemisphere \(\tau ^{-}\subset \hat{\sigma }\) with ∂τ−=σ. We call a singular sphere \(\sigma \subset \hat{\sigma }\) maximal, if it is not contained in a singular sphere in \(\hat{\sigma }\) of strictly larger dimension.
Note that if \(\hat{F}\) is not Morse, then \(\hat{\sigma }\) itself is singular.
Corollary 4.8
Let X be a locally compact CAT(0) space. Let \(\hat{F}\) be a periodic (n+1)-flat in X with \(\partial_{T}\hat{F}=\hat {\sigma }\). Then the set of maximal singular spheres in \(\hat{\sigma }\) is finite.
Proof
Proposition 4.6 implies that a maximal singular sphere \(\sigma \subset \hat{\sigma }\) has to be isolated in the following sense. If (σk) is a sequence of singular spheres which converges to a sphere σ∞⊂σ, then we must have σk⊂σ for almost all \(k\in \mathbb{N}\). This yields the claim. □
Recall that a point in \(\hat{\sigma }\) is regular, if it has a Tits-neighborhood which is entirely contained in \(\hat{\sigma }\). Otherwise we call it singular.
Lemma 4.9
Every singular point in \(\hat{\sigma }\) is contained in a singular sphere.
Proof
If \(\xi \in \hat{\sigma }\) is a singular point, then there exists a sequence (ξk) in \(\partial_{T}X\setminus \hat{\sigma }\) with ξk→ξ. By Lemma 4.4, there exists a flat half-plane H orthogonal to \(\hat{F}\) and with ξ∈∂∞H. Set \(\tau ^{+}_{1}:=\partial_{\infty}H\). If \(\partial \tau ^{+}_{1}\) is not singular, then by the Lune Lemma (Lemma 2.1) we can find a half-circle \(\tau ^{-}_{1}\subset \hat{\sigma }\) with \(\partial \tau ^{+}_{1}=\partial \tau ^{-}_{1}\) such that \(\tau ^{+}_{1}\) and \(\tau ^{-}_{1}\) span a 2-dimensional spherical lune λ2. By Corollary 4.5, there exists a 2-dimensional round hemisphere \(\tau ^{+}_{2}\subset \partial_{T}X\) with \(\tau ^{+}_{2}\cap \hat{\sigma }=\partial \tau ^{+}_{2}\) and \(\tau ^{-}_{1}\subset \partial \tau ^{+}_{2}\), in particular \(\xi \in \partial \tau ^{+}_{2}\). If \(\partial \tau ^{+}_{2}\) is not singular, we repeat the argument to produce a 3-dimensional round hemisphere \(\tau ^{+}_{3}\subset \partial_{T}X\) with \(\xi \in \partial \tau ^{+}_{3}\). After at most (n−1) steps we end up with a singular sphere that contains the point ξ. □
Now we obtain the following refinement of Theorem D.
Theorem 4.10
Let X be a locally compact CAT(0) space. Suppose that X contains a periodic Morse flat F. Then the complement in ∂TF of the set of regular points is a finite union of singular spheres of positive codimension.
Proof
Immediate from Corollary 4.8 and Lemma 4.9. □
Proof of Theorem A
Let F⊂X be a periodic n-flat. By Theorem D, σ=∂TF contains a dense subset of spherical points. Then, by Theorem 3.2, ∂TX is a spherical join or a spherical building. Since X is geodesically complete and locally compact, the claim follows from Theorem 2.8 and [KL97, Proposition 2.3.7]. □
Proof of Corollary C
By Theorem A, Lemma 3.6 and Lemma 4.3, we can produce a product decomposition of X as claimed. Since every such decomposition induces a corresponding join decomposition of ∂TX. The uniqueness statement follows from the uniqueness of such join decompositions [Lyt05, Corollary 1.2] or alternatively from [FL08, Theorem 1.1]. □
Notes
For cube complexes, the statement of Higher Rank Rigidity takes the form of a splitting result, since neither irreducible symmetric spaces nor irreducible Euclidean buildings are cube complexes.
See [OR22] for a definition of rank for CAT(0) spaces whose isometry groups satisfy the duality condition.
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Acknowledgements
I would like to use this opportunity to thank Bernhard Leeb and Alexander Lytchak for pointing out several mistakes in a late version of this article. Their insightful comments led to substantial improvements. I also want to thank Bruce Kleiner for several inspiring discussions. In addition, I want to thank the anonymous referee for helpful comments. I was supported by DFG grant SPP 2026.
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Stadler, S. CAT(0) Spaces of Higher Rank I. Geom. Funct. Anal. 34, 512–528 (2024). https://doi.org/10.1007/s00039-024-00661-2
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DOI: https://doi.org/10.1007/s00039-024-00661-2