CAT(0) Spaces of Higher Rank I

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.

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];

• for a 3-dimensional simplicial complex with a piecewise Euclidean CAT(0) metric and a geometric group action [BB00];

• for a finite dimensional CAT(0) cube complex with a geometric group action [CS11];¹

• 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];

• 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

$$ X=\mathbb{R}^{j}\times X_{1}\times \cdots \times X_{k}\times Y_{1}\times \cdots \times Y_{l}\times M_{1}\times \cdots \times M_{m} $$

where X_a are rank 1 spaces, Y_b are irreducible Euclidean buildings of higher rank and M_c are irreducible Riemannian symmetric spaces of higher rank.

Note that the assumptions above imply that the factors X_a are not only rank 1 but have discrete Tits boundaries. Thus if X admits a cocompact group action, then every X_a 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.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 B_r(x) and $\bar{B}_{r}(x)$ the open and closed r-ball around x, respectively. Similarly, N_r(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 (Σ_xX,∠), 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:

$$ \dim (X)= \sup _{x\in X} \{ \dim (\Sigma _{x}X) +1 \}. $$

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 B_r(ξ)⊂Z is isometric to an r-ball in S^k. 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 Σ_ξZ has an antipode in Σ_ξσ≅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 Σ_zZ is geodesically complete. Then for every pair of directions v,w∈Σ_zZ there exists an antipode $\hat{w}\in \Sigma _{z} Z$ of w, such that

$$ \angle _{z}(\hat{w},v)=\pi -\angle _{z}(w,v). $$

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 π_C: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

$$ P(Y)\cong Y\times CS(Y) $$

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

$$ \partial_{T}(Y_{1}\cap Y_{2})=\partial_{T}Y_{1}\cap \partial_{T}Y_{2}. $$

For points ξ∈∂_∞X and x∈X we denote the Busemann function centered at ξ based at x by b_ξ,x. If ρ:[0,∞)→X denotes the geodesic ray asymptotic to ξ and with ρ(0)=x, then

$$ b_{\xi ,x}(y)=\lim _{t\to \infty}(|y,\rho (t)|-t). $$

It is a 1-Lipschitz convex function whose negative gradient at a point y∈X is given by log_y(ξ). We denote the horoball centered at a point ξ∈∂_∞X and based at the point x∈X by

$$ HB(\xi ,x):=b^{-1}_{\xi ,x}((-\infty ,0]). $$

It is a closed convex subset with

$$ \partial_{T}HB(\xi ,x)=\bar{B}_{\frac{\pi}{2}}(\xi )\subset \partial_{T}X. $$

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.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

$$ P(\sigma )=P(F) $$

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 c_k in X with $\partial_{\infty}c_{k}=\{\hat{\xi},\xi _{k}\}$ and ξ_k→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 c_k extending the geodesic ray $\rho (k)\hat{\xi}$ and such that

$$ \angle _{\rho (k)}(\hat{\xi},\xi )=\pi -\angle _{\rho (k)}(\xi ,\xi _{k}) $$

where ξ_k:=c_k(+∞). Now for every p∈oξ we have

$$ \angle _{p}(\xi _{k},\xi )\leq \angle _{\rho (k)}(\xi _{k},\xi )=\pi - \angle _{\rho (k)}(\xi ,\hat{\xi})\to 0. $$

Since $\angle _{\rho (k)}(\xi ,\hat{\xi})\to |\xi ,\hat{\xi}|=\pi$. As p∈oξ was arbitrary, this shows ξ_k→ξ in ∂_∞X. Indeed, by local compactness, we may assume ξ_k→ξ′ after passing to a subsequence. In particular, ∠_x(ξ_k,ξ′)→0 for all x∈X. Now if ξ≠ξ′, then there exists q∈oξ such that ∠_q(ξ,ξ′)>0. But

$$ \angle _{q}(\xi ,\xi ')\leq \angle _{q}(\xi ,\xi _{k})+\angle _{q}( \xi _{k},\xi ')\to 0. $$

 □

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 B_s(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 (c_k) in X with $\partial_{\infty}c_{k}=\{\xi ,\hat{\xi}_{k}\}$ and $\hat{\xi}_{k}\to \hat{\xi}$. By assumption, there exists n-flats F_k with c_k⊂F_k. 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,

$$ \pi =|\xi ,\hat{\xi}|\leq |\xi ,\eta |+|\eta ,\hat{\xi}|\leq s+\liminf \limits _{k\to \infty}|\eta ,\hat{\xi}_{k}|=\pi , $$

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 B_s(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∈B_s(ξ), we must have B_s(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(∂_TX_j)=n_j−1, j=1,2, and every complete geodesic in X_j is contained in an n_j-flat.

Proof

It is clear that the rank is additive. So let c_j be a complete geodesic in X_j 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 F_j⊂X_j such that $\tilde{F}\subset F_{1}\times F_{2}$. In particular, dim(F_j)=n_j and c_j⊂F_j. □

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 B_ϵ(ξ)⊂U′. If σ⊂∂_TX denotes a round (n−1)-sphere with ξ∈σ, then σ∩B_ϵ(ξ)=B_ϵ(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

$$ H\cap F=\partial H\ \text{ and }\ \angle (H,F)\geq \frac{\pi}{2} $$

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 (ξ_k) 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 x_k∈oξ_k such that |x_k,F|=R. Denote by $\bar{x}_{k}\in F$ the nearest point to x_k. Choose isometries γ_k∈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 γ_k(oξ_k,x_k)→(c_∞,x_∞) with respect to pointed Hausdorff topology. The complete geodesic c_∞ is asymptotic to ξ since |γ_kξ_k,ξ|=|ξ_k,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 (H_k) of dimension d≤n, all orthogonal to $\hat{F}$, such that their boundary flats h_k:=∂H_k 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 σ_k=∂_∞h_k converges to a round sphere $\sigma \subset \hat{\sigma }$ and for every $k\in \mathbb{N}$ the convex hull of σ_k 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 (H_k,p_k)→(H_∞,p_∞), where p_k∈F_k. 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 ζ_k 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 p_kζ_k converge to p_∞ζ_∞, we find a sequence t_k→∞ and points x_k∈p_kζ_k such that $b_{\zeta _{\infty},p_{\infty}}(x_{k})\leq -t_{k}$. The geodesic ray p_kζ_k lies in H_k and therefore x_k lies in P(h_k). Thus HB(ζ_∞,x_k)∩P(h_k) is a non-empty closed convex set and we have

$$ \partial_{\infty}(HB(\zeta _{\infty},x_{k})\cap P(h_{k}))=\bar{B}_{\frac{\pi}{2}}( \zeta _{\infty})\cap \partial_{\infty}P(h_{k}). $$

As $\hat{F}\subset P(h_{k})$, we see ∂_∞h_∞⊂∂_∞(HB(ζ_∞,x_k)∩P(h_k)). From Lemma 2.7 we infer that there is a (d−1)-flat E_k⊂HB(ζ_∞,x_k)∩P(h_k) with ∂_∞E_k=∂_∞h_∞. Since the flats h_k are pairwise non-parallel and E_k⊂P(h_k), we see that there is a flat F_k in P(h_k) of dimension at least d which contains E_k and such that ∂_∞h_k⊂∂_∞F_k. Note that $b_{\zeta _{\infty},p_{\infty}}$ is bounded above on E_k by −t_k since E_k⊂HB(ζ_∞,x_k). 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 F_k, then $|F_{k},F'_{k}|_{H}\geq t_{k}$. In particular, F_k and $F'_{k}$ bound a flat strip of dimension ≥d+1, orthogonal to $\hat{F}$ and of width at least t_k. Since $\hat{F}$ is periodic and t_k→∞, 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 E_k⊂P(h_k) spans a flat F_k⊂P(h_k) of dimension $\hat{d}$ if and only if the convex hull of the spheres ∂_∞E_k=∂_∞h_∞ and ∂_∞h_k 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 (σ_k) is a sequence of singular spheres which converges to a sphere σ_∞⊂σ, then we must have σ_k⊂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 (ξ_k) in $\partial_{T}X\setminus \hat{\sigma }$ with ξ_k→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]. □