Injective Hulls of Infinite Totally Split-Decomposable Metric Spaces ^†

Abstract

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. For this class, we provide a characterization for the injective hull to be combinatorially equivalent to a CAT(0) cube complex. Intermediate results include the generalization of the decomposition theory introduced by Bandelt and Dress in 1992 as well as results on the tight span of totally split-decomposable metric spaces proved by Huber, Koolen, and Moulton in 2006. Next, using results of Lang from 2013, we obtain proper actions on CAT(0) cube complexes for finitely generated groups endowed with a totally split-decomposable word metric and for which the associated splits satisfy a simple combinatorial property. In the case of Gromov hyperbolic groups, the obtained action is both proper aand co-compact. Finally, we obtain as an application that injective hulls of odd cycles are cell complexes isomorphic to CAT(0) cube complexes.

1. Introduction

In 1992, Bandelt and Dress (cf. [1]) introduced a decomposition theory for finite metric spaces which is canonical, that is, it is the only one which is in a sense compatible with Isbell’s injective hull (see [2] for a gentle introduction). Our first goal is to extend the canonical decomposition theory to the class of infinite metric spaces with an integer-valued totally split-decomposable metric and possessing an injective hull with the structure of a polyhedral complex. For this class, we then provide necessary and sufficient conditions for the injective hull to be combinatorially equivalent to a CAT(0) cube complex.

This work builds a bridge between different worlds; on the one side, our results extend the work of Bandelt, Dress, and Buneman [3] on finite metric spaces and the correspondence with canonical cell complexes as well as the later extension of those ideas by Huber, Koolen, and Moulton [4,5,6]. Our results show that the decomposition theory can be successfully applied to infinite metric spaces. Moreover, the present work establishes connections with the results of Lang [7], which lie in another area of mathematics dealing with applications to geometric group theory. The group actions that are obtained as a consequence of our results offer new tools in the theory of actions on CAT(0) cube complexes and ultimately aim at contributing to advance the general classification of groups, a long-standing and major goal of mathematical research. Finally, we illustrate our results by providing a concise and elegant characterization of the injective hull of odd cycles. In this way, we recover as a special case results from Suter which were obtained using completely different techniques, namely, Hasse diagrams of Young lattices [8].

The basic definitions of the canonical decomposition theory of Bandelt and Dress do not need to be modified to suit our more general situation. A split (also called cut) $S=\{A,B\}$ of a set X is a pair of non-empty subsets of X such that $A\cap B=\varnothing$ and $X=A\cup B$, or in other words, $X=A\bigsqcup B$. For $x\in X$, we denote by $S\left(x\right)$ the element of S that contains x. The split (pseudo-)metric associated with S is then a pseudometric ${\delta }_S$ on X such that

$${\delta }_S(x,y):=\left\{\begin{array}{cc}1\hfill & \mathrm{if}S\left(x\right)\ne S\left(y\right),\hfill \\ 0\hfill & \mathrm{if}S\left(x\right)=S\left(y\right).\hfill \end{array}\right.$$

For a pseudometric d on X, we refer to $S=\{A,B\}$ as a d-split (of X) if the isolation index provided by

$$\begin{array}{c}{a}_S^d:={\displaystyle \frac{1}{2}}\underset{\begin{array}{c}a,a^{\prime }\in \mathcal{A}\\ b,b^{\prime }\in \mathcal{B}\end{array}}{inf}\left[max\left\{\begin{array}{cc}\hfill d(a,b)& \hfill +d(a^{\prime },b^{\prime })\\ \hfill d(a,a^{\prime })& \hfill +d(a,b^{\prime })\\ \hfill d(b,b^{\prime })& \hfill +d(a,a^{\prime })\end{array}\right\}-d(a,a^{\prime })-d(b,b^{\prime })\right]\end{array}$$

(1)

satisfies ${\alpha }_S^d>0$. The pseudometric d is called totally split-decomposable if $d={\sum }_{S\in \mathcal{S}}{\alpha }_S^d{\delta }_S$, where $\mathcal{S}$ is the set of all d-splits. A split subsystem $\overline{\mathcal{S}}\subset \mathcal{S}$ is called octahedral if and only if there is a partition of X into a disjoint union of six non-empty sets $X=Y_1^1\bigsqcup Y_1^{-1}\bigsqcup Y_2^1\bigsqcup Y_2^{-1}\bigsqcup Y_3^1\bigsqcup Y_3^{-1}$ such that $\overline{\mathcal{S}}$ consists of the following four splits:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1& :=\{Y_1^1\bigsqcup Y_2^1\bigsqcup Y_3^1,Y_1^{-1}\bigsqcup Y_2^{-1}\bigsqcup Y_3^{-1}\}\hfill \\ \hfill S_2& :=\{Y_1^1\bigsqcup Y_2^1\bigsqcup Y_3^{-1},Y_1^{-1}\bigsqcup Y_2^{-1}\bigsqcup Y_3^1\}\hfill \\ \hfill S_3& :=\{Y_1^1\bigsqcup Y_2^{-1}\bigsqcup Y_3^1,Y_1^{-1}\bigsqcup Y_2^1\bigsqcup Y_3^{-1}\}\hfill \\ \hfill S_4& :=\{Y_1^1\bigsqcup Y_2^{-1}\bigsqcup Y_3^{-1},Y_1^{-1}\bigsqcup Y_2^1\bigsqcup Y_3^1\},\hfill \end{array}\end{array}$$

(2)

while $\mathcal{S}$ is called octahedral-free if it does not contain any octahedral split subsystem. Two splits $S:=\{A,B\}$ and $S^{\prime }:=\{A^{\prime },B^{\prime }\}$ are said to be compatible if $A^{\prime }\subset A$ (and thus $B\subset B^{\prime }$), or alternatively if $A\subset A^{\prime }$ (and thus $B^{\prime }\subset B$).

We provide an outline of the structure of Isbell’s injective hull and present conditions under which it has the structure of a polyhedral complex, following [7]. Given a pseudometric space $(X,d)$, let us consider the vector space ${\mathbb{R}}^X$ of real-valued functions on X and $\Delta (X,d):=\{f\in {\mathbb{R}}^X:f\left(x\right)+f\left(y\right)\ge d(x,y)\mathrm{for}\mathrm{all}x,y\in X\}$. We refer to $f\in \Delta (X,d)$ as extremal if there is no $g\le f$ in $\Delta (X,d)$ distinct from f. The set $\mathrm{E}(X,d)$ of extremal functions is equivalently provided by

$$\mathrm{E}(X,d)=\left\{f\in {\mathbb{R}}^X:f\left(x\right)={sup}_{y\in X}(d(x,y)-f\left(y\right))\mathrm{for}\mathrm{all}x\in X\right\}.$$

(3)

In order to be able to describe the structure of $\mathrm{E}(X,d)$ further, it is possible to assign the undirected graph with vertex set X and edge set $A\left(f\right):=\left\{\{x,y\}:x,y\in X\mathrm{and}f\left(x\right)+f\left(y\right)=d(x,y)\right\}$ to every $f\in \mathrm{E}(X,d)$, allowing self-loops $\{x,x\}$ which correspond to zeros of f. Furthermore, we let ${\mathrm{E}}^{\prime }(X,d):=\left\{f\in \Delta (X,d):\bigcup A\left(f\right)=X\right\}$. Note that if $f\in {\mathrm{E}}^{\prime }(X,d)$, then the graph $(X,A(f\left)\right)$ has no isolated vertices (although it may be disconnected). A set A of unordered pairs of (possibly equal) points in X is called admissible if there exists an $f\in {\mathrm{E}}^{\prime }(X,d)$ with $A\left(f\right)=A$, and we denote by $\mathcal{A}\left(X\right)$ the collection of admissible sets. To every $A\in \mathcal{A}\left(X\right)$, we associate the affine subspace $H\left(A\right)$ of ${\mathbb{R}}^X$ provided by $H\left(A\right):=\{g\in {\mathbb{R}}^X:A\subset A\left(g\right)\}=\{g\in {\mathbb{R}}^X:g\left(x\right)+g\left(y\right)=d(x,y)\mathrm{for}\mathrm{all}x,y\in A\}$. We define the rank of A by $\mathrm{rank}\left(A\right):=\dim \left(H\right(A\left)\right)\in \mathbb{N}\cup \{0,\infty \}$, which is provided by the number of bipartite components or even A-components of $(X,A)$. As an example, if $d_x:X\to \mathbb{R}$ denotes the distance function to $x\in X$ i.e., $d_x\left(y\right):=d(x,y)$, then $\mathrm{rank}\left(A\left(d_x\right)\right)=0$. We also have another example to show that the rank can be infinite; indeed, if $(S^1,d)$ denotes the unit circle endowed with the intrinsic geodesic metric and if $f\in \mathrm{E}(S^1,d)$ denotes the function that is constantly equal to $\frac{\mathrm{diam}\left(S^1\right)}{2}=\frac{\pi }{2}$, then it is possible to verify that $\mathrm{rank}\left(A\right(f\left)\right)=\infty$. See [9] for a discussion of related examples.

If $(X,d)$ is a finite metric space, then $\mathrm{E}(X,d)$ is a finite polyhedral complex; if $(X,d)$ is infinite, then we say that $(X,d)$ satisfies the local rank condition (LRC) if and only if for every $f\in \mathrm{E}(X,d)$ there exist $\epsilon ,N>0$ such that for all $g\in {\mathrm{E}}^{\prime }(X,d)$ with ${\parallel f-g\parallel }_{\infty }<ϵ$ we have $\mathrm{rank}\left(A\right(g\left)\right)\le N$. Recall (cf. [7], Theorem 4.5) that if $(X,d)$ is a metric space with an integer-valued metric and satisfying the (LRC), then $\mathrm{E}(X,d)={\mathrm{E}}^{\prime }(X,d)$; in this case, let $P\left(A\right):={\mathrm{E}}^{\prime }(X,d)\cap H\left(A\right)=\mathrm{E}(X,d)\cap H\left(A\right)=\Delta (X,d)\cap H\left(A\right)$. Then, the family ${\left\{P\left(A\right)\right\}}_{A\in \mathcal{A}\left(X\right)}$ defines a polyhedral structure on $\mathrm{E}(X,d)$; in particular, $P\left(A^{\prime }\right)$ is a face of $P\left(A\right)$ if and only if $A\subset A^{\prime }$. This defines a canonical locally finite dimensional polyhedral structure on $\mathrm{E}(X,d)$. In the case where d is totally split-decomposable, our goal is to provide necessary and sufficient conditions ensuring that $\mathrm{E}(X,d)$ is combinatorially equivalent to a $\mathrm{CAT}\left(0\right)$ cube complex. Accordingly, we have the following theorem.

Theorem 1.

Let $(X,d)$ be a metric space with an integer-valued totally split-decomposable metric satisfying the local rank condition and let $\mathcal{S}$ be the set of all d-splits; then, the following are equivalent:

(i) 

$\mathcal{S}$ does not contain any octahedral split subsystem $\overline{\mathcal{S}}$, satisfying the requirement that for every $S=\{A,B\}\in \mathcal{S}\setminus \overline{\mathcal{S}}$ there is $S^{\prime }:=\{A^{\prime },B^{\prime }\}\in \overline{\mathcal{S}}$ such that S and $S^{\prime }$ are compatible.

(ii) 

Each cell of $\mathrm{E}(X,d)$ is a parallelotope.

• If (i) or equivalently (ii) in Theorem 1 holds, then there is a CAT(0) cube complex $\mathrm{K}(X,d)$ and a canonical bijective cell complex isomorphism $\sigma :\mathrm{E}(X,d)\to \mathrm{K}(X,d)$ mapping cells affinely to cells.

For general facts regarding injective hulls, we refer to [7]. Injective hulls can be characterized in several different ways. In the sequelae, the injective hull refers to Isbell’s injective hull construction $(X,d)\mapsto \mathrm{E}(X,d)$. The difference between two elements of $\mathrm{E}(X,d)$ has a finite ${∥·∥}_{\infty }$-norm, and $\mathrm{E}(X,d)$ is endowed with the metric $d_{\infty }(f,g):={∥f-g∥}_{\infty }$. It is easy to see that for $f\in \mathrm{E}(X,d)$, if $d(x,x^{\prime })=0$, then $f\left(x\right)=f\left(x^{\prime }\right)$; hence, if $(X,d)$ is a pseudometric space and $(Y,d^{\prime })$ is the associated metric space obtained by collapsing every maximal set of diameter zero to a single point, then $\mathrm{E}(X,d)$ and $\mathrm{E}(Y,d^{\prime })$ are isometric. Accordingly, statements involving the injective hull will be stated for metric spaces instead of for pseudometric spaces.

We use the word parallelotope for a Minkowski sum of a finite collection of linearly independent closed segments (for instance, see [10]). Whenever condition (i) in Theorem 1 holds, we say that the family of all d-splits of X has no compatibly octahedral decomposition. If the diameters of the cells of $\mathrm{E}(X,d)$ are uniformly bounded, then $\sigma$ in Theorem 1 can be chosen to be bi-Lipschitz. For a metric space $(X,d)$, let $I(x,y):=\{z\in X:d(x,z)+d(z,y)=d(x,y\left)\right\}$ denote the interval between x and y, and for $x,v\in X$ let the cone determined by the directed pair $(x,v)$ be provided by $C(x,v):=\{y\in X:v\in I(x,y\left)\right\}$. For a subset B of X, we denote by $\mathcal{C}\left(B\right)$ the set of all pointed cones $(v,C(x,v\left)\right)$ with $v\in B$ and $x\in X$. In addition, $(X,d)$ is called discretely geodesic if the metric is integer-valued and for every pair of points $x,y\in X$ there exists an isometric embedding $\gamma :\{0,1,\dots ,d(x,y\left)\right\}\to X$ such that $\gamma \left(0\right)=x$ and $\gamma \left(d\right(x,y\left)\right)=y$. Moreover, we say that a discretely geodesic metric space X has β-stable intervals for some constant $\beta \ge 0$ if, for every triple of points $x,y,y^{\prime }\in X$ such that $d(y,y^{\prime })=1$, we have $d_H(I(x,y),I(x,y^{\prime }))\le \beta$, where $d_H$ denotes the Hausdorff distance in X.

Among its other features, the injective hull has applications to geometric group theory. Let $\Gamma$ be a finitely generated group and G a finite generating set, and let $\Gamma$ be equipped with the word metric $d_G$ with respect to the alphabet $G\cup G^{-1}$. Note that the metric $d_G$ is integer-valued.

Theorem 2.

Let $(\Gamma ,d_G)$ be a metric space where Γ is a finitely generated group and G a finite generating set, and let $d_G$ denote the word metric defined on Γ with respect to the alphabet $G\cup G^{-1}$. Assume that $d_G$ is totally split-decomposable and that (i) or equivalently (ii) in Theorem 1 holds. Then, the following hold:

(i) 

If $(\Gamma ,d_G)$ has β-stable intervals, then there is a proper action of Γ (a CAT(0) cube complex $\mathrm{K}(X,d)$$\mathrm{K}(\Gamma ,d_G)$) provided by

$$(x,y)\mapsto (\sigma \circ {\overline{L}}_x\circ {\sigma }^{-1})\left(y\right).$$

(ii) 

If $(\Gamma ,d_G)$ is δ-hyperbolic, then the action of Γ on $\mathrm{K}(\Gamma ,d_G)$ is also co-compact.

To prove Theorem 1, and consequently Theorem 2, we need to decompose any pseudometric d on a set X in a way that is coherent with the structure of $\mathrm{E}(X,d)$. The isolation index of a pair $S:=\{A,B\}$ of non-empty subsets with respect to a pseudometric d on X is the non-negative number ${\alpha }_S^d$ (equivalently ${\alpha }_{\{A,B\}}^d$, or simply ${\alpha }_S$) provided by (1). We then refer to a pseudometric $d_0$ on X as split-prime if ${\alpha }_S^{d_0}=0$ for any split S of X. Note that per Lemma 1, for any integer-valued pseudometric there are only finitely many d-splits separating any pair of points.

Theorem 3.

Let $(X,d)$ be a pseudometric space with an integer-valued pseudometric, let ${\mathcal{S}}_X$ be the set of all splits of X, and let $\mathcal{S}$ be the set all d-splits. Letting

$$\left\{\begin{array}{cc}{\lambda }_S\in [0,{\alpha }_S^d]\hfill & \mathrm{if}S\in \mathcal{S},\hfill \\ {\lambda }_S=0\hfill & \mathrm{if}S\in {\mathcal{S}}_X\setminus \mathcal{S},\hfill \end{array}\right.$$

$\tilde{d}:=d-{\sum }_{S\in \mathcal{S}}{\lambda }_S{\delta }_S$ is a pseudometric such that for every split $S\in {\mathcal{S}}_X$ we have ${\alpha }_S^{\tilde{d}}={\alpha }_S^d-{\lambda }_S$; in particular, there is a split-prime pseudometric $d_0$ such that

$$d=d_0+\sum _{S\in \mathcal{S}}{\alpha }_S^d{\delta }_S.$$

The decomposition provided by Theorem 3 can be characterized uniquely in a corollary to Theorem 6 which relates to the structure of general cellular graphs; see [11] (Section 8.3).

Corollary 1.

Let $(X,d)$ be a metric space with an integer-valued metric satisfying the LRC. Let $\mathcal{S}$ be the family of all d-splits of X such that $d=d_0+{\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$, and let ${\lambda }_S\in [0,{\alpha }_S]$ for every $S\in \mathcal{S}$. Then, setting $d_1:=d-{\sum }_{S\in \mathcal{S}}{\lambda }_S{\delta }_S$, we have

$$\mathrm{E}(X,d)\subset {\mathbb{R}}^X\cap \left(\mathrm{E}(X,d_1)+\sum _{S\in \mathcal{S}}{\lambda }_S\mathrm{E}(X,{\delta }_S)\right).$$

(4)

Moreover, for any split $S=\{A,B\}$ of X and any ${\lambda }_S>0$ such that $d=d_1+{\lambda }_S{\delta }_S$, if $\mathrm{E}(X,d)\subset \mathrm{E}(X,d_1)+{\lambda }_S\mathrm{E}(X,{\delta }_S)$, then the following hold:

(i) 

S is a d-split of X.

(ii) 

${\lambda }_S\le {\alpha }_S$.

We conclude with a discussion of the future research directions that can build on the present work. The results presented here echo the discussion in [12], where interest in infinite metric spaces and CAT(0) complexes were highlighted. The conclusions of our work suggest that other ideas which are originally developed for finite metric spaces, for instance in phylogenetics, can be successfully generalized to the infinite case. Subsequently, such correspondence can be leveraged to infer results about abstract metric spaces in order to provide links to other areas of mathematics with a novel perspective, such as geometric group theory. Finally, splits in metric spaces are not only relevant in the case of tight spans; there has also been recent work on links with Vietoris–Rips complexes [13] with applications to topological data analysis (TDA), which is an exciting direction for further research.

2. Decomposition Theory

It is easy to see that $\mathrm{E}(X,d)$ as defined at the beginning of the introduction is a subset of ${\Delta }_1(X,d):=\{f\in \Delta (X,d):f\mathrm{is}1-\mathrm{Lipschitz}\}$. Note that a function $f\in {\mathbb{R}}^X$ belongs to ${\Delta }_1(X,d)$ if and only if

$$\parallel f-d_x{\parallel }_{\infty }=f\left(x\right)\mathrm{for}\mathrm{all}x\in X,$$

(5)

where $d_x:X\to \mathbb{R}$ denotes the distance function to $x\in X$, i.e., $d_x\left(y\right):=d(x,y)$. The metric $d_{\infty }(f,g):={\parallel f-g\parallel }_{\infty }$ on ${\Delta }_1(X,d)$ is thus well-defined and $\mathrm{E}(X,d)$ is equipped with the induced metric. We have the canonical isometric mapping $\mathrm{e}:(X,d)\to \mathrm{E}(X,d)$, provided by $\mathrm{e}\left(x\right)=d_x$. Let $(X,d)$ be any pseudometric space. A partial split $S=\{A,B\}$ of X is a pair of non-empty subsets of X such that $A\cap B=\varnothing$. If $X=A\cup B$ also holds, then $S=\{A,B\}$ is a split of X. A partial d-split is a partial split $S=\{A,B\}$ for which ${\alpha }_S^d>0$. Note now that if $(X,d)$ is a pseudometric space with integer-valued pseudometric, then for any split S we have

$${\alpha }_S\in [0,\infty )\cap {\displaystyle \frac{1}{2}}\mathbb{Z}.$$

(6)

The proof of the next lemma is very similar to the proof of its finite analogue, except that we cannot use induction on the set of all d-splits [14] (IV.2 Theorem 2.3). Instead, we use the fact that for every $\epsilon >0$ there are $\{a,a^{\prime }\}\subset A_0$ and $\{b,b^{\prime }\}\subset B_0$, such that ${\alpha }_{\{A_0,B_0\}}+\epsilon \ge {\alpha }_{\{\{a,a^{\prime }\},\{b,b^{\prime }\}\}}$ (if $x=y$, we still use $\{x,y\}$ to denote $\left\{x\right\}$). Moreover, if $Y\subset X$ satisfies $Y\cap A_0,Y\cap B_0\ne \varnothing$, then we have ${\alpha }_{\{A_0,B_0\}}^d\le {\alpha }_{\{Y\cap A_0,Y\cap B_0\}}^{{d|}_{Y\times Y}}$.

Theorem 4.

Let $(X,d)$ be a pseudometric space such that, for every $x,y\in X$, there are only finitely many distinct d-splits S satisfying $S\left(x\right)\ne S\left(y\right)$. Moreover, let $\{A_0,B_0\}$ be a partial d-split. Then,

$$\sum \left\{{\alpha }_{\{A,B\}}:\{A,B\}\mathrm{is}\mathrm{a}d-splitextending\{A_0,B_0\}\right\}\le {\alpha }_{\{A_0,B_0\}}.$$

The next lemma enables us to use several arguments of [1] by examining the pointwise behavior of pseudometrics. The approach of [15], which generalizes [1], has a global nature which causes obstructions to any direct adaptation to the case with $\left|X\right|=\infty$.

Lemma 1.

Let $(X,d)$ be a pseudometric space with an integer-valued pseudometric. For every $x,y\in X$, there are at most $2d(x,y)$ distinct d-splits S satisfying $S\left(x\right)\ne S\left(y\right)$.

Proof. 

Assume by contradiction that we can find $m:=2d(x,y)+1$ distinct d-splits $S_i$ such that $S_i\left(x\right)\ne S_i\left(y\right)$. For every $1\le i<j\le m$, choose $z_{ij}\in X$ such that either $z_{ij}\in S_i\left(x\right)\cap S_j\left(y\right)$ or $z_{ij}\in S_i\left(y\right)\cap S_j\left(x\right)$. Furthermore, setting $Z:=\{x,y\}\cup \{z_{ij}:1\le i<j\le m\}$, we have $S_i\left(x\right)\cap Z\ne S_j\left(x\right)\cap Z$ if $i\ne j$; hence

$$d(x,y)+\frac{1}{2}\le \sum _{i\in \{1,\dots ,m\}}{\alpha }_{S_i}^d\le \sum _{i\in \{1,\dots ,m\}}{\alpha }_{\{S_i\left(x\right)\cap Z,S_i\left(y\right)\cap Z\}}^{{d|}_{Z\times Z}},$$

and per the special case ([1], Theorem 1) of Theorem 4 applied to the partial split $A_0:=\left\{x\right\}$ and $B_0:=\left\{y\right\}$ of the finite set Z, the right-hand side is less than or equal to ${\alpha }_{\left\{\right\{x\},\{y\left\}\right\}}^{{d|}_{Z\times Z}}=d(x,y)$, which is a contradiction. This concludes the proof. □

It is possible to prove the next lemma by a direct modification of ([1], Theorem 2).

Lemma 2.

Let $(X,d)$ be a pseudometric space with an integer-valued pseudometric. Let $\mathcal{S}$ be the set of all d-splits of X and let $\tilde{\mathcal{S}}\subset \mathcal{S}$ be any finite subset. If ${\lambda }_S\in (0,{\alpha }_S^d]$ for every $S\in \tilde{\mathcal{S}}$ and ${\lambda }_S:=0$ for every other split, then $\tilde{d}:=d-{\sum }_{S\in \tilde{\mathcal{S}}}{\lambda }_S{\delta }_S$ is a pseudometric such that for every split S of X we have

$${\alpha }_S^{\tilde{d}}={\alpha }_S^d-{\lambda }_S.$$

The next definition is taken from [1], where it is introduced for finite metric spaces. The same definition extends to the infinite case without modification.

Definition 1.

We refer to a collection $\mathcal{S}$ of splits of a set X as weakly compatible if there are no four points $\{x_0,x_1,x_2,x_3\}\subset X$ and three splits $\{S_1,S_2,S_3\}\subset \mathcal{S}$ such that for any $i,j\in \{1,2,3\}$ one has $S_i\left(x_0\right)=S_i\left(x_j\right)⟺i=j$.

It follows immediately from the above definition that the collection of all d-splits with respect to any pseudometric d on X is weakly compatible. This is part of the next theorem.

Theorem 5.

Let $(X,d)$ denote any pseudometric space with an integer-valued metric; then, the d-splits of X are weakly compatible. Conversely, let ${\mathcal{S}}_0$ denote any collection of weakly compatible splits of X and choose for each $S\in {\mathcal{S}}_0$ some ${\lambda }_S\in (0,\infty )$ such that

$$d:=\sum _{S\in {\mathcal{S}}_0}{\lambda }_S{\delta }_S:X\times X\to \mathbb{Z}\cap [0,\infty ).$$

It follows that ${\mathcal{S}}_0$ is the set of all d-splits and that for each $S\in {\mathcal{S}}_0$, the isolation index ${\alpha }_S:={\alpha }_S^d$ coincides with ${\lambda }_S$.

The proof of Theorem 5 is a close analogue to the proof of ([1], Theorem 3); see [14]. The assumptions in the next lemma ensure that $\mathrm{E}(X,d)$ is a polyhedral complex. We denote by ${\mathrm{\Sigma }}^0\left(\mathrm{E}(X,d)\right)$ the set of vertices of the polyhedral complex $\mathrm{E}(X,d)$; equivalently, it is the set of all functions $f\in \mathrm{E}(X,d)$ such that $\mathrm{rank}\left(A\right(f\left)\right)=0$. The idea of considering this set of functions is inspired by [15] (Lemma 2.2). Both Lemma 5 and Theorem 6 highlight differences that arise when extending the theory of finite metric spaces to infinite ones.

Lemma 3.

Let $(X,d)$ be a metric space with an integer-valued metric satisfying the LRC, and let $f\in {\mathrm{\Sigma }}^0\left(\mathrm{E}(X,d)\right)$. Let ${\mathcal{S}}_{<\infty }$ be any finite subset of the set $\mathcal{S}$ of all d-splits of X. If we pick ${\lambda }_S\in [0,{\alpha }_S^d]$ for every $S\in {\mathcal{S}}_{<\infty }$, then we have functions $f_S\in \mathrm{E}(X,{\delta }_S)$ such that

$$f-\sum _{S\in {\mathcal{S}}_{<\infty }}{\lambda }_S{f}_S\in \Delta \left(X,d-\sum _{S\in {\mathcal{S}}_{<\infty }}{\lambda }_S{\delta }_S\right).$$

Proof. 

Let $S:=\{A,B\}\in {\mathcal{S}}_{<\infty }$, where ${\mathcal{S}}_{<\infty }$ is any finite subset $\mathcal{S}$. Note that $f\in {\mathrm{\Sigma }}^0\left(\mathrm{E}(X,d)\right)$ means that f is a vertex of the polyhedral complex $\mathrm{E}(X,d)$, which is equivalent to $\mathrm{rank}\left(A\right(f\left)\right)=0$ by definition. Thus, $A\left(f\right)$ is in particular not bipartite, meaning that there are $a,a^{\prime }\in X$ such that $\{a,a^{\prime }\}\in A\left(f\right)$ and either $a,a^{\prime }\in A$ or $a,a^{\prime }\in B$. Assume without loss of generality that $a,a^{\prime }\in A$. Note also that if there are $b,b^{\prime }\in B$ such that we have $\{b,b^{\prime }\}\in A\left(f\right)$, then ${\alpha }_S^d=0$, which contradicts our assumption. Hence, for any $b,b^{\prime }\in B$, we have $\{b,b^{\prime }\}\notin A\left(f\right)$. Now, setting

$$f_S\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\in A,\hfill \\ 1\hfill & \mathrm{if}x\in B,\hfill \end{array}\right.$$

it readily follows that $f^{\left(1\right)}:=f-{\lambda }_S{f}_S$ and $d^{\left(1\right)}:=d-{\lambda }_S{f}_S$ satisfy $f^{\left(1\right)}\in \Delta \left(X,d^{\left(1\right)}\right)$. Furthermore, note that if $\{x,y\}\in A^d\left(f\right)$, then either $x,y\in A$ or $x\in A$ and $y\in B$. In both cases, it is evident that $\{x,y\}\in A^{d^{\left(1\right)}}\left(f^{\left(1\right)}\right)$ (noting that $A^{d^{\left(1\right)}}\left(f^{\left(1\right)}\right)$ refers to those pairs $\{x,y\}$ satisfying $f^{\left(1\right)}\left(x\right)+f^{\left(1\right)}\left(y\right)=d^{\left(1\right)}(x,y)$). Thus,

$$A^d\left(f\right)\subset A^{d^{\left(1\right)}}\left(f^{\left(1\right)}\right);$$

(7)

hence, in particular, $f^{\left(1\right)}\in {\mathrm{\Sigma }}^0\left(\mathrm{E}(X,d^{\left(1\right)})\right)$. For $S^{\prime }:=\{A^{\prime },B^{\prime }\}\in \mathcal{S}\setminus \left\{S\right\}$, we can proceed as above and find $c,c^{\prime }\in A^{\prime }$ such that $\{c,c^{\prime }\}\in A^{d^{\left(1\right)}}\left(f^{\left(1\right)}\right)$. Now, by Theorem 3, it follows that ${\alpha }_{S^{\prime }}^{d^{\left(1\right)}}={\alpha }_{S^{\prime }}^d>0$. This implies that for any $e,e^{\prime }\in B^{\prime }$, we have $\{e,e^{\prime }\}\notin A^{d^{\left(1\right)}}\left(f^{\left(1\right)}\right)$. Now, setting

$$f_{S^{\prime }}^{\left(1\right)}\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\in A^{\prime },\hfill \\ 1\hfill & \mathrm{if}x\in B^{\prime }\hfill \end{array}\right.$$

we obtain $f^{\left(1\right)}-{\lambda }_{S^{\prime }}{f}_{S^{\prime }}^{\left(1\right)}\in \Delta \left(X,d^{\left(1\right)}-{\lambda }_{S^{\prime }}{\delta }_{S^{\prime }}\right)$, as before. Per (7), it follows that $f_{S^{\prime }}=f_{S^{\prime }}^{\left(1\right)}$. It then follows as a result that

$$f-{\lambda }_S{f}_S-{\lambda }_{S^{\prime }}{f}_{S^{\prime }}=f^{\left(1\right)}-{\lambda }_{S^{\prime }}{f}_{S^{\prime }}^{\left(1\right)}\in \Delta \left(X,d^{\left(1\right)}-{\lambda }_{S^{\prime }}{\delta }_{S^{\prime }}\right)=\Delta \left(X,d^{\left(2\right)}\right),$$

where $d^{\left(2\right)}=d-{\lambda }_S{\delta }_S-{\lambda }_{S^{\prime }}{\delta }_{S^{\prime }}$. Proceeding by induction, we get the desired result. □

For an infinite split system $\mathcal{S}$, we generally have $\Delta (X,d_0)+{\sum }_{S\in \mathcal{S}}{\alpha }_S\Delta (X,{\delta }_S)\subset {[0,\infty ]}^X$ and cannot replace ${[0,\infty ]}^X$ by ${\mathbb{R}}^X$.

Theorem 6.

Let $(X,d)$ be a metric space with an integer-valued metric satisfying the LRC. Let $\mathcal{S}$ be the family of all d-splits of X such that $d=d_0+{\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ and let ${\lambda }_S\in [0,{\alpha }_S]$ for every $S\in \mathcal{S}$, setting $d_1:=d-{\sum }_{S\in \mathcal{S}}{\lambda }_S{\delta }_S$. Then,

$$\Delta (X,d)={\mathbb{R}}^X\cap \left(\Delta (X,d_1)+\sum _{S\in \mathcal{S}}{\lambda }_S\Delta (X,{\delta }_S)\right).$$

(8)

Proof. 

Let $\mathcal{S}$ be the set of all d-splits and let $S\in \mathcal{S}$. As in the proof of Lemma 3, $f\in {\mathrm{\Sigma }}^0\left(\mathrm{E}(X,d)\right)$ means that f is a vertex of the polyhedral complex $\mathrm{E}(X,d)$, which by definition is equivalent to $\mathrm{rank}\left(A\right(f\left)\right)=0$. Thus, $A\left(f\right)$ is not bipartite, which means that there are $a,a^{\prime }\in X$ such that $\{a,a^{\prime }\}\in A\left(f\right)$. Without loss of generality, we assume that $a,a^{\prime }\in A$. Then, for any $b,b^{\prime }\in B$, we have $\{b,b^{\prime }\}\notin A\left(f\right)$. Now, setting

$$f_S\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}x\in A,\hfill \\ 1\hfill & \mathrm{if}x\in B,\hfill \end{array}\right.$$

we first show that for every $x\in X$, we have ${\sum }_{S\in \mathcal{S}}{\lambda }_S{f}_S\left(x\right)\ne \infty$. Note that for every $x\in X$ there exists $x^{\prime }\in X$ such that $\{x,x^{\prime }\}\in A^d\left(f\right)$. Furthermore, we have

$$d(x,x^{\prime })=d_0(x,x^{\prime })+\sum _{\begin{array}{c}S\in \mathcal{S}\\ S\left(x\right)\ne S\left(x^{\prime }\right)\end{array}}{\alpha }_S^d{\delta }_S(x,x^{\prime }).$$

Because ${\alpha }_S^d\in [0,\infty )\cap {\displaystyle \frac{1}{2}}\mathbb{Z}$, we deduce that the set

$${\mathcal{S}}_{x{x}^{\prime }}:=\{S\in \mathcal{S}:S\left(x\right)\ne S\left(x^{\prime }\right)\}$$

is finite. Moreover, for every $S:=\{A,B\}\in \mathcal{S}\setminus {\mathcal{S}}_{x{x}^{\prime }}$, because $\{x,x^{\prime }\}\in A^d\left(f\right)$, by definition we have $x,x^{\prime }\in A$ and $f_S\left(x\right)=0=f_S\left(x^{\prime }\right)$. It then follows that ${\sum }_{S\in \mathcal{S}}{\lambda }_S{f}_S\in {\mathbb{R}}^X$ as well as that

$$f_1:=f-\sum _{S\in \mathcal{S}}{\lambda }_S{f}_S\in {\mathbb{R}}^X.$$

For $d_1:=d-{\sum }_{S\in \mathcal{S}}{\lambda }_S{\delta }_S$, it now remains to show that $f_1\in \Delta (X,d_1)$. For every $x,y\in X$, there are $x^{\prime },y^{\prime }\in X$ such that $\{x,x^{\prime }\},\{y,y^{\prime }\}\in A^d\left(f\right)$. Because for every $S\in {\mathcal{S}}_{xy}$ we have $f_S\left(x\right)+f_S\left(y\right)=1$, it follows that ${\mathcal{S}}_{xy}\subset {\mathcal{S}}_{x{x}^{\prime }}\cup {\mathcal{S}}_{y{y}^{\prime }}=:{\mathcal{S}}_{<\infty }$, where $|{\mathcal{S}}_{<\infty }|<\infty$. By Lemma 3 and setting $d_1^{<\infty }:=d-{\sum }_{S\in {\mathcal{S}}_{<\infty }}{\lambda }_S{\delta }_S$, it follows that

$$f_1^{<\infty }:=f-\sum _{S\in {\mathcal{S}}_{<\infty }}{\lambda }_S{f}_S\in \Delta \left(X,d_1^{<\infty }\right),$$

and thus that $f_1\left(x\right)+f_1\left(y\right)=f_1^{<\infty }\left(x\right)+f_1^{<\infty }\left(y\right)\ge d_1(x,y)$. This shows that ${\mathrm{\Sigma }}^0\left(\mathrm{E}(X,d)\right)\subset \left(\Delta (X,d_1)+{\sum }_{S\in \mathcal{S}}{\lambda }_S\Delta (X,{\delta }_S)\right)$. Now, using the fact that $(X,d)$ satisfies the LRC, we can consider for each cell of $\mathrm{E}(X,d)$ all finite convex combinations of its vertices, and by convexity of $\Delta (X,d_1)$ and $\Delta (X,{\delta }_S)$, for every $S\in \mathcal{S}$ we can deduce that $\mathrm{E}(X,d)\subset \left(\Delta (X,d_1)+{\sum }_{S\in \mathcal{S}}{\lambda }_S\Delta (X,{\delta }_S)\right)$. Finally, adding ${[0,\infty )}^X$ on both sides and intersecting with ${\mathbb{R}}^X$, we get

$$\Delta (X,d)\subset {\mathbb{R}}^X\cap \left(\Delta (X,d_1)+\sum _{S\in \mathcal{S}}{\lambda }_S\Delta (X,{\delta }_S)\right).$$

Because the other inclusion is easy to see, we obtain the desired result. □

Hence, we obtain the first part of the statement of Corollary 1. By virtue of Remark 1 below, the proof of the second part of the statement of the corollary is similar to the proof of the corresponding assertion for the set $\Delta (X,d)$ in ([1], (Theorem 7).

Remark 1.

Properties (4) and (8) are equivalent. It is shown in the proof of Theorem 6 that (4) implies (8). To see that the other implication holds, remember that by [7] (Proposition 3.1) there is a 1-Lipschitz map $p:\Delta (X,d)\to \mathrm{E}(X,d)$ such that one has $g:=p\left(f\right)\le f$ for every $f\in \Delta (X,d)$. From (4), we obtain a decomposition of g as $g:=g_1+{\sum }_{S\in \mathcal{S}}{\lambda }_S{g}_S$. Moreover, $f-g\in {[0,\infty )}^X$; hence,

$$f=g_1+(f-g)+\sum _{S\in \mathcal{S}}{\lambda }_S{g}_S\in {\mathbb{R}}^X\cap \left(\Delta (X,d_1)+\sum _{S\in \mathcal{S}}{\lambda }_S\Delta (X,{\delta }_S)\right),$$

which is the desired result.

3. The Buneman Complex and Related Topics

If $\mathcal{S}$ is a split system on a set X and $\alpha :\mathcal{S}\to (0,\infty )$ is any map $S\mapsto {\alpha }_S$, then the pair $(\mathcal{S},\alpha )$ is called a split system pair (of X). If $\mathcal{S}$ is weakly compatible, as in Definition 1, then $(\mathcal{S},\alpha )$ is called a weakly compatible split system pair. Now, let $\mathcal{S}$ be a weakly compatible split system on a pseudometric space with an integer-valued pseudometric $(X,d)$ and assume that $d={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$. Per Theorem 5, $\mathcal{S}$ is the set of all d-splits of X; thus, d totally split-decomposable. The weakly compatible split system pair $(\mathcal{S},\alpha )$ is called the split system pair associated with $(X,d)$. Unless otherwise stated, this $(\mathcal{S},\alpha )$ is the split system pair that we refer to in the rest of the discussion when considering a totally split-decomposable pseudometric space $(X,d)$. We want to stress that the sets $\mathcal{S}$ and X are generally infinite.

Definition 2.

We refer to $\mathrm{K}$ as acell complexif K is a subset of a real vector space endowed with a family of convex subsets ${\left\{C_i\right\}}_{i\in I}$ of $\mathrm{K}$ such that the collection ${\Delta }_{\mathrm{K}}:={\left\{C_i\right\}}_{i\in I}$ verifies that $C_i\cap C_j\in {\Delta }_{\mathrm{K}}\cup \{\varnothing \}$ and ${\bigcup }_i{C}_i=\mathrm{K}$ for any $C_i,C_j\in {\Delta }_{\mathrm{K}}$. The sets $C_i$ are called the cells of $\mathrm{K}$, and the dimension of $C_i$ is the dimension of its affine hull, which in general is infinite.

Let $(\mathcal{S},\alpha )$ be any split system pair on a set X, and consider

$$U\left(\mathcal{S}\right):=\{A\subset X:\mathrm{there}\mathrm{is}S\in \mathcal{S}\mathrm{such}\mathrm{that}A\in S\}.$$

For a map $\mu :U\left(\mathcal{S}\right)\to [0,\infty )$, we can write $\mathrm{supp}\left(\mu \right):=\{A\in U(\mathcal{S}):\mu (A)>0\}$. If $A\subset X$, then we denote the complement $X\setminus A$ by $\overline{A}$. For a given $A\in U\left(\mathcal{S}\right)$, we denote the associated split $\{A,\overline{A}\}\in \mathcal{S}$ by $S_A$. We define the following hypercube:

$$\begin{array}{c}\hfill H(S,\alpha ):=\left\{\mu :\mathcal{U}\left(S\right)\to [0,\infty )|\begin{array}{c}\mathrm{for}\mathrm{all}A\in \mathcal{U}\left(S\right),\mathrm{one}\mathrm{has}\mu \left(A\right)\ge 0\hfill \\ \mathrm{and}\mu \left(A\right)+\mu \left(\tilde{A}\right)=a_S^{\alpha }/2\hfill \end{array}\right\}\end{array}$$

which is in general infinitely dimensional. Here, $H(\mathcal{S},\alpha )$ has a natural cell complex structure; cells are sets of the form

$$\left[\mu \right]:=\{{\mu }^{\prime }\in H(\mathcal{S},\alpha ):\mathrm{supp}\left({\mu }^{\prime }\right)\subset \mathrm{supp}\left(\mu \right)\},$$

(9)

where $\mu \in H(\mathcal{S},\alpha )$. The cells of $H(\mathcal{S},\alpha )$ are (possibly) infinite-dimensional hypercubes. The Buneman complex is the subcomplex of $H(\mathcal{S},\alpha )$ provided by

$$\begin{array}{c}\hfill B(S,\alpha ):=\left\{\mu \in H(S,\alpha );\mathrm{if}A,B\in supp\left(\mu \right)\mathrm{and}A\cup B=X,\mathrm{then}A\cap B=\varnothing \right\}.\end{array}$$

Next, we define

$$\begin{array}{c}\hfill \overline{T}(\mathcal{S},\alpha ):=\left\{\mu \in H(\mathcal{S},\alpha )|\begin{array}{c}\mathrm{if}{\left\{A_i\right\}}_{i\in I}\subset supp\left(\mu \right)\mathrm{and}\bigcup _{i\in I}{A}_i=X,\hfill \\ \mathrm{then}\bigcap _{i\in I}{A}_i=\varnothing \hfill \end{array}\right\}.\end{array}$$

It is easy to see that $\overline{T}(\mathcal{S},\alpha )$ is a subcomplex of $B(\mathcal{S},\alpha )$, as in the finite case ([16], Section 4). For $x\in X$, the map ${\varphi }_x:U\left(\mathcal{S}\right)\to [0,\infty )$ is defined as

$${\varphi }_x\left(A\right):=\left\{\begin{array}{cc}\frac{1}{2}{\alpha }_{S_A}\hfill & \mathrm{if}x\notin A,\hfill \\ 0\hfill & \mathrm{if}x\in A.\hfill \end{array}\right.$$

Furthermore, let $d_1:{\mathbb{R}}^{U\left(\mathcal{S}\right)}\times {\mathbb{R}}^{U\left(\mathcal{S}\right)}\to {[0,\infty ]}^X$ be provided by

$$(\mu ,\psi )\mapsto \sum _{A\in U\left(\mathcal{S}\right)}|\mu \left(A\right)-\psi \left(A\right)|.$$

The map $\kappa :{\mathbb{R}}^{U\left(\mathcal{S}\right)}\to {[0,\infty ]}^X$ is provided by $\mu \mapsto \kappa \left(\mu \right)$, where $\kappa \left(\mu \right)\left(x\right)=d_1(\mu ,{\varphi }_x)$ for $x\in X$. As a direct application of the above definitions, we obtain the following lemma.

Lemma 4.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X and assume that $d:={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ defines a pseudometric on X. Then, the following hold:

(i) 

For every $x,y\in X$, we have $d_1({\varphi }_x,{\varphi }_y)=d(x,y)$.

(ii) 

For every $x\in X$, we have ${\varphi }_x\in \overline{T}(\mathcal{S},\alpha )\subset B(\mathcal{S},\alpha )$.

(iii) 

${sup}_{x\in X}|\kappa \left(\mu \right)\left(x\right)-\kappa \left(\varphi \right)\left(x\right)|\le d_1(\mu ,\varphi )$, where each side might be infinite.

Under the assumptions of Lemma 4, for $x\in X$ and $S=\{A,\overline{A}\}\in \mathcal{S}$, let $S\left(x\right):=A$ if $x\in A$ and $S\left(x\right):=\overline{A}$ if $x\in \overline{A}$. For a further $y\in X$ and $\psi :\mathcal{S}\to \mathbb{R}$, a direct calculation shows that

$${\displaystyle \frac{1}{2}}[\kappa \left(\psi \right)\left(x\right)+\kappa \left(\psi \right)\left(y\right)-d(x,y)]\ge \sum _{\begin{array}{c}S\in \mathcal{S}\\ S\left(x\right)=S\left(y\right)\end{array}}\left[\left|\psi \left(S\right(x\left)\right)\right|+\left|\psi \left(\overline{S\left(x\right)}\right)-\frac{{\alpha }_S}{2}\right|\right]$$

(10)

and that equality holds if $\psi \in H(\mathcal{S},\alpha )$.

Lemma 5.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X and assume that $d:={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ defines a pseudometric on X. Then, for every $\mu \in H(\mathcal{S},\alpha )$, the following are equivalent:

(i) 

$\kappa \left(\mu \right)\in {\mathrm{E}}^{\prime }(X,d)$.

(ii) 

$\mu \in \overline{T}(\mathcal{S},\alpha )$.

Proof. 

Consider $\mu \in H(\mathcal{S},\alpha )$. We first show that $\left(ii\right)$ implies $\left(i\right)$. Let $\mu \in H(\mathcal{S},\alpha )$, and assume that $\kappa \left(\mu \right)\notin {\mathrm{E}}^{\prime }(X,d)$. For $x,y\in X$, we have $\kappa \left(\mu \right)\left(x\right)+\kappa \left(\mu \right)\left(y\right)=d_1(\mu ,{\varphi }_x)+d_1(\mu ,{\varphi }_y)\ge d_1({\varphi }_x,{\varphi }_y)=d(x,y)$; thus, $\kappa \left(\mu \right)\in {[0,\infty ]}^X$. By our contradiction assumption, there is $x\in X$ such that for every $y\in X$ we have $\kappa \left(\mu \right)\left(x\right)+\kappa \left(\mu \right)\left(y\right)>d(x,y)$, where the left-hand side is possibly infinite. Because $\mu \in H(\mathcal{S},\alpha )$, equality holds in (10); hence, there is $S_y\in \mathcal{S}$ such that $S_y\left(x\right)=S_y\left(y\right)$ and $\mu \left(S_y\left(x\right)\right)>0$. Therefore, $X={\bigcup }_{y\in X}{S}_y\left(x\right)$ where ${\left\{S_y\right\}}_{y\in X}\subset \mathrm{supp}\left(\mu \right)$. Moreover, $x\in {\bigcap }_{y\in X}{S}_y\left(x\right)\ne \varnothing$. It follows that $\mu \notin \overline{T}(\mathcal{S},\alpha )$. This shows that $\left(ii\right)$ implies $\left(i\right)$.

To show the other implication, assume that $\kappa \left(\mu \right)\in {\mathrm{E}}^{\prime }(X,d)$. For every $x\in X$, there is $w\in X$ such that $\kappa \left(\mu \right)\left(x\right)+\kappa \left(\mu \right)\left(w\right)=d(x,w)$. Per (10), we have the following for any $S\in \mathcal{S}$:

$$\mathrm{if}S\left(x\right)=S\left(w\right),\mathrm{then}\mu \left(S\right(x\left)\right)=0=\mu \left(S\right(w\left)\right).$$

(11)

Note that for any $A_i\in \mathrm{supp}\left(\mu \right)$, there exists by definition $S_i\in \mathcal{S}$ as well as $x_i\in X$ such that $A_i=S_i\left(x_i\right)$. Now, if $X={\bigcup }_{i\in I}{A}_i={\bigcup }_{i\in I}{S}_i\left(x_i\right)$ and assuming by contradiction that ${\bigcap }_{i\in I}{S}_i\left(x_i\right)\ne \varnothing$, we can pick an arbitrary $z\in {\bigcap }_{i\in I}{S}_i\left(x_i\right)$. Then, for every $i\in I$ we have $S_i\left(x_i\right)=S_i\left(z\right)$, and thus $X={\bigcup }_{i\in I}{S}_i\left(z\right)$. It is now clear that the existence of z contradicts (11). Indeed, for any $y\in X$, there is $S_j\in {\left\{S_i\right\}}_{i\in I}\subset \mathcal{S}$ such that $y\in S_j\left(z\right)$, and hence

$$S_j\left(y\right)=S_j\left(z\right).$$

(12)

However,

$$S_j\left(z\right)=S_j\left(x_j\right)=A_j\in \mathrm{supp}\left(\mu \right)$$

(13)

and y can be chosen so that $\kappa \left(\mu \right)\left(z\right)+\kappa \left(\mu \right)\left(y\right)=d(z,y)$. Thus, by (11) and (12), we have $\mu \left(S_j\left(y\right)\right)=0=\mu \left(S_j\left(z\right)\right)$ which is a contradiction to (13). This finishes the proof. □

Using Corollary 1, Lemma 5 and proceeding similarly to ([17], Theorem 3.1), we can immediately deduce the next lemma; see [14] (IV.3, 3.4 Lemma). Note that under the assumptions of Lemma 6, we have $\mathrm{E}(X,d)={\mathrm{E}}^{\prime }(X,d)$.

Lemma 6.

Let $(X,d)$ be a totally split-decomposable metric space with an integer-valued metric satisfying the LRC. Then, ${\kappa |}_{\overline{T}(\mathcal{S},\alpha )}:\overline{T}(\mathcal{S},\alpha )\to \mathrm{E}(X,d)$ is surjective.

For a map $\varphi :\mathcal{S}\to [0,\infty )$, let $\mathcal{S}\left(\varphi \right):=\{S\in \mathcal{S}:S\subset \mathrm{supp}(\varphi \left)\right\}$. Define for a cell $\left[\varphi \right]$ of $\overline{T}(\mathcal{S},\alpha )$ and $x\in X$ the map ${\gamma }_{\left[\varphi \right]}^x:U\left(\mathcal{S}\right)\to [0,\infty )$ provided by

$${\gamma }_{\left[\varphi \right]}^x\left(A\right):=\left\{\begin{array}{cc}{\varphi }_x\left(A\right)\hfill & \mathrm{if}A\in U\left(\mathcal{S}\right(\varphi \left)\right),\hfill \\ \varphi \left(A\right)\hfill & \mathrm{if}A\in U(\mathcal{S}\setminus \mathcal{S}(\varphi \left)\right).\hfill \end{array}\right.$$

Note that if $A\in U(\mathcal{S}\setminus \mathcal{S}(\varphi \left)\right)$ and $\psi \in \left[\varphi \right]$, then $\psi \left(A\right)=\varphi \left(A\right)={\gamma }_{\left[\varphi \right]}^x\left(A\right)$. Therefore, $\mathrm{supp}\left({\gamma }_{\left[\varphi \right]}^x\right)\subset \mathrm{supp}\left(\varphi \right)$; hence, ${\gamma }_{\left[\varphi \right]}^x\in \left[\varphi \right]$ by (9).

Definition 3.

Let $i:(X,d_X)\to (Y,d_Y)$ be an isometric map of pseudometric spaces. We say that $Z\subset Y$ is X-gated (for i and with respect to $d_Y$) if and only if for every $x\in X$ there is $y_x\in Z$ such that for every $z\in Z$ we have $d_Y(i\left(x\right),z)=d_Y(i\left(x\right),y_x)+d_Y(y_x,z)$.

From Lemma 5, it follows that the restriction of $d_1$ to $\overline{T}(\mathcal{S},\alpha )$ defines a metric. The proof of the next lemma proceeds as the proof of ([4], Lemma 3.1); see [14] (IV.3, 3.6 Lemma).

Lemma 7.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X and assume that $d:={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ defines a pseudometric on X. Then, every cell $\left[\varphi \right]$ of $\overline{T}(\mathcal{S},\alpha )$ is X-gated with respect to the restriction of $d_1$ to $\overline{T}(\mathcal{S},\alpha )$.

By virtue of Lemma 5, we also obtain the next lemma.

Lemma 8.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X and assume that $d:={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ defines a pseudometric on X. Then, for every $\varphi \in \overline{T}(\mathcal{S},\alpha )$, the split system $\mathcal{S}\left(\varphi \right)\subset \mathcal{S}$ is antipodal, which means that for any $x\in X$ there is $y\in X$ such that

$$foreveryS\in \mathcal{S}\left(\varphi \right),onehasS\left(x\right)\ne S\left(y\right).$$

(14)

For $x,y\in X$, if $d(x,y)=\kappa \left(\varphi \right)\left(x\right)+\kappa \left(\varphi \right)\left(y\right)$, then x and y satisfy (14).

Proof. 

Let $\varphi \in \overline{T}(\mathcal{S},\alpha )$. By Lemma 5, we have $\kappa \left(\varphi \right)\in {\mathrm{E}}^{\prime }(X,d)$. Thus, for any $x\in X$, there is $y\in X$ such that $d(x,y)=\kappa \left(\varphi \right)\left(x\right)+\kappa \left(\varphi \right)\left(y\right)$, which can be rewritten as

$$\sum _{S\in \mathcal{S}}{\alpha }_S{\delta }_S(x,y)=d_1(\varphi ,{\varphi }_x)+d_1(\varphi ,{\varphi }_y).$$

(15)

It is easy to see that for every $S\in \mathcal{S}$, we have

$${\alpha }_S{\delta }_S(x,y)=\sum _{A\in S}|{\varphi }_x\left(A\right)-{\varphi }_y\left(A\right)|\le \sum _{A\in S}\left[|{\varphi }_x\left(A\right)-\varphi \left(A\right)|+|\varphi \left(A\right)-{\varphi }_y\left(A\right)|\right],$$

which together with (15) implies

$${\alpha }_S{\delta }_S(x,y)=\sum _{A\in S}\left[|{\varphi }_x\left(A\right)-\varphi \left(A\right)|+|\varphi \left(A\right)-{\varphi }_y\left(A\right)|\right].$$

(16)

Assume now that there is $S\in \mathcal{S}\left(\varphi \right)$ such that $S\left(x\right)=S\left(y\right)$; then, per (16) we have $0={\alpha }_S{\delta }_S(x,y)={\sum }_{A\in S}\left[|{\varphi }_x\left(A\right)-\varphi \left(A\right)|+|\varphi \left(A\right)-{\varphi }_y\left(A\right)|\right]=4\varphi \left(S\left(x\right)\right)$, which implies $S\left(x\right)\notin \mathrm{supp}\left(\varphi \right)$ and thus $S\notin \mathcal{S}\left(\varphi \right)$, which is a contradiction. This finishes the proof. □

By Lemma 7, every cell $\left[\varphi \right]$ of $\overline{T}(\mathcal{S},\alpha )$ is X-gated. Let $(\Gamma \left(\left[\varphi \right]\right),d_1)$ denote the set of all X-gates of $\left[\varphi \right]$ endowed with the restriction of $d_1$. A pseudometric space $(X,d)$ is called antipodal if there exists an involution $\sigma :X\to X$ such that for every $x,y\in X$ one has $d(x,\sigma (x\left)\right)=d(x,y)+d(y,\sigma (x\left)\right)$. With Lemma 8, the proof of the next lemma is easily seen to be similar to that of ([4] Lemma 4.2); see [14] (IV.3, 3.8 Lemma).

Lemma 9.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X and assume that $d:={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ defines a pseudometric on X. Then, for every cell $\left[\varphi \right]$ of $\overline{T}(\mathcal{S},\alpha )$, the metric space $(\Gamma \left(\left[\varphi \right]\right),d_1)$ is antipodal.

For $\psi \in \overline{T}(\mathcal{S},\alpha )$, we know from Lemma 5 that $\kappa \left(\psi \right)\in {\mathrm{E}}^{\prime }(X,d)$. Recalling from the introduction that $H\left(A\right):=\{g\in {\mathbb{R}}^X:A\subset A\left(g\right)\}$ and setting $A:=A\left(\kappa \right(\psi \left)\right))$, we let $\left[\kappa \left(\psi \right)\right]:=H\left(A\right)\cap {\mathrm{E}}^{\prime }(X,d)$. If $(X,d)$ has an integer-valued metric and satisfies the LRC, $\mathrm{E}(X,d)$ is a cell complex in which all cells are of this form.

Lemma 10.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X and assume that $d:={\sum }_{S\in \mathcal{S}}{\alpha }_S{\delta }_S$ defines a pseudometric on X. Then, for every cell $\left[\psi \right]$ of $\overline{T}(\mathcal{S},\alpha )$, one has $\kappa \left(\right[\psi \left]\right)\subset \left[\kappa \right(\psi \left)\right]$.

Lemma 10 (see [14], IV.3, 3.9 Lemma) follows easily from a direct computation, i.e., it is possible to show that for each $\{x,y\}$ such that $\kappa \left(\psi \right)\left(x\right)+\kappa \left(\psi \right)\left(y\right)=d(x,y)$ we have the following equality for every $\mu \in \left[\psi \right]$:

$$d_1({\varphi }_x,{\varphi }_y)=d_1({\varphi }_x,{\gamma }_{\left[\psi \right]}^x)+d_1({\gamma }_{\left[\psi \right]}^x,\mu )+d_1(\mu ,{\gamma }_{\left[\psi \right]}^y)+d_1({\gamma }_{\left[\psi \right]}^y,{\varphi }_y)$$

(17)

which implies $\kappa \left(\mu \right)\left(x\right)+\kappa \left(\mu \right)\left(y\right)=d(x,y)$.

Remark 2.

For $f\in {\mathbb{R}}^X$, let $H\left(A\left(f\right)\right):={\bigcap }_{\{x,y\}\in A\left(f\right)}\partial H_{\{x,y\}}$, where $\partial H_{\{x,y\}}:=\{g\in {\mathbb{R}}^X:g\left(x\right)+g\left(y\right)=d(x,y)\}$. Set ${\mathcal{S}}_{xy}:=\{S\in \mathcal{S}:S\left(x\right)\ne S\left(y\right)\}$ and $Z_{\{x,y\}}:=\partial H_{\{x,y\}}\cap {\sum }_{S\in \mathcal{S}}{\alpha }_S\mathrm{E}(X,{\delta }_S)$, then:

(i) 

Assume that $(X,d)$ has an integer-valued metric, is totally split-decomposable, and satisfies the LRC. In the proof Lemma 11 below, we only require that every cell $\left[f\right]$ of $\mathrm{E}(X,d)$ can be written as

$$\left[f\right]=\sum _{S\in {\mathcal{S}}_{\left[f\right]}}{\alpha }_S\mathrm{E}(X,{\delta }_S)+\sum _{S\in \mathcal{S}\setminus {\mathcal{S}}_{\left[f\right]}}{\alpha }_S{p}_S,$$

(18)

where for each $S\in \mathcal{S}\setminus {\mathcal{S}}_{\left[f\right]}$ one has $p_S\in {\{0,1\}}^X$. To see that this holds, note first that $\left[f\right]=H\left(A\left(f\right)\right)\cap \mathrm{E}(X,d)=H\left(A\left(f\right)\right)\cap {\sum }_{S\in \mathcal{S}}{\alpha }_S\mathrm{E}(X,{\delta }_S)$ per Theorem 6; then, it is easy to see from the definition of the sets $\mathrm{E}(X,{\delta }_S)$ and the fact that we have a decomposition $f={\sum }_{S\in \mathcal{S}}{\alpha }_S{f}_S$ with $f_S\in \mathrm{E}(X,{\delta }_S)$ for every $S\in \mathcal{S}$ that $Z_{\{x,y\}}={\sum }_{S\in {\mathcal{S}}_{xy}}{\alpha }_S\mathrm{E}(X,{\delta }_S)+{\sum }_{S\in \mathcal{S}\setminus {\mathcal{S}}_{xy}}{\alpha }_S{p}_S$ where $p_S{|}_{S\left(x\right)}\equiv 0$ and $p_S{|}_{\overline{S\left(x\right)}}\equiv 1$. In addition, for every ${\sum }_{S\in \mathcal{S}}{\alpha }_S{g}_S\in Z_{\{x,y\}}$, we have $g_S=p_S$ for each $S\in \mathcal{S}$ satisfying $S\left(x\right)=S\left(y\right)$.

(ii) 

It is not difficult to see that if $(X,d)$ is as in (i) and if every cell $\left[f\right]$ of $\mathrm{E}(X,d)$ is a combinatorial hypercube, then the representation in (18) verifies

$$k:=\dim \left(\left[f\right]\right)=|{\mathcal{S}}_{\left[f\right]}|.$$

(19)

This can easily be proved by induction using the fact that every cell can be written as the Minkowski sum of all its edges incident to a single vertex. If $f={\sum }_{S\in \mathcal{S}}{\alpha }_S{f}_S$ is as usual and if ${\mathcal{S}}_f:=\{S\in \mathcal{S}:f_S\left(X\right)\subset (0,1)\}$, then it is easy to see that $\left[f\right]={\sum }_{S\in {\mathcal{S}}_f}{\alpha }_S\mathrm{E}(X,{\delta }_S)+{\sum }_{S\in \mathcal{S}\setminus {\mathcal{S}}_f}{\alpha }_S{p}_S$; in particular, ${\mathcal{S}}_{\left[f\right]}={\mathcal{S}}_f$.

(iii) 

For $(X,d)$, as in (ii), let us define $\lambda :\mathrm{E}(X,d)\to \overline{T}(\mathcal{S},\alpha )$ by the assignment $f\mapsto {\psi }_f$, where ${\psi }_f$ is defined for every $S=\{A,\overline{A}\}$ as well as for arbitrarily chosen $x\in A$ and $y\in \overline{A}$ by setting

$${\psi }_f\left(A\right):={\displaystyle \frac{{\alpha }_S}{2}}{f}_S\left(x\right)and{\psi }_f\left(\overline{A}\right):={\displaystyle \frac{{\alpha }_S}{2}}{f}_S\left(y\right).$$

This definition depends on a choice of a representation ${\sum }_{S\in \mathcal{S}}{\alpha }_S{f}_S$ for f, and in general this choice is not unique. We denote by $\overline{f}$ an arbitrarily chosen element of $\left[f\right]$ such that $\left[\overline{f}\right]=\left[f\right]$ and that ${\mathcal{S}}_{\overline{f}}$ is maximal among the elements of $\left[f\right]$. Furthermore, note that we always have $\kappa \circ \lambda ={\mathrm{id}}_{\mathrm{E}(X,d)}$. It follows that κ is surjective. In general, $\lambda \circ \kappa \ne {\mathrm{id}}_{\overline{T}(\mathcal{S},\alpha )}$; however, if every cell $\left[f\right]$ of $\mathrm{E}(X,d)$ is a combinatorial hypercube, then the map $\kappa :\overline{T}(\mathcal{S},\alpha )\to \mathrm{E}(X,d)$ defines a bijection as well as an isomorphism of cell complexes.

The proof of the next lemma is an easy consequence of Remark 2; see [14] (IV.3, 3.15 Lemma).

Lemma 11.

Let $(X,d)$ be a totally split-decomposable metric space with an integer-valued metric which satisfies the LRC. For every cell $\left[f\right]$ of $\mathrm{E}(X,d)$, if $\overline{f}\in \left[f\right]$ and ${\psi }_{\overline{f}}\in \overline{T}(S,\alpha )$ are defined as in (iii) of Remark 2, then one has $\left[\kappa \left({\psi }_{\overline{f}}\right)\right]\subset \kappa \left(\left[{\psi }_{\overline{f}}\right]\right)$.

Recall that there is a canonical isometric embedding $\mathrm{e}:X\to \mathrm{E}(X,d)$ provided by $x\mapsto (d_x:y\mapsto d(x,y))$ in which $\mathrm{E}(X,d)$ is endowed with the metric $d_{\infty }(f,g):={\parallel f-g\parallel }_{\infty }$. Assume that $(X,d)$ satisfies the assumptions of Lemma 11. We say that $\mathrm{E}(X,d)$ is cell-decomposable if every cell C of $\mathrm{E}(X,d)$ is X-gated (cf. Definition 3). Now, a direct computation shows that $\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^x\right)$ is a gate for $d_x$ in $\left[\overline{f}\right]=\left[f\right]$. Thus, we have the following lemma; see also [14] (IV.3, 3.16 Lemma).

Lemma 12.

Let $(X,d)$ be a totally split-decomposable metric space with an integer-valued metric satisfying the LRC. Then, $\mathrm{E}(X,d)$ is cell-decomposable.

Let $(G\left(\left[f\right]\right),d_{\infty })$ denote the set of all X-gates of $\left[f\right]$ endowed with the restriction of the metric $d_{\infty }$. We denote by ${\gamma }_{\left[f\right]}^x$ the gate of $d_x$ in $\left[f\right]$. It is easy to see that the proof of ([6], Theorem 1.1) directly generalizes to the case where $\left|X\right|=\infty$ as long as ${\mathrm{E}}^{\prime }(X,d)=\mathrm{E}(X,d)$ and using an extension of the Mazur-Ulam Theorem [18]. Hence, we have the next theorem; see [14] (IV.3, 3.17 Theorem).

Theorem 7.

Let $(X,d)$ be a metric space with an integer-valued metric satisfying the LRC. If $f\in \mathrm{E}(X,d)$ is such that $\left[f\right]$ is X-gated, then the following hold:

(i) 

$(G\left(\left[f\right]\right),d_{\infty })$ is a finite antipodal metric space.

(ii) 

The map $\mathrm{\Phi }:(\left[f\right],d_{\infty })\to \mathrm{E}(G\left(\left[f\right]\right),d_{\infty })$ provided by

$$g\mapsto \left({\gamma }_{\left[f\right]}^x\mapsto g\left(x\right)-{\gamma }_{\left[f\right]}^x\left(x\right)\right)$$

(20)

is a bijective isometry as well as an isomorphism of polytopes.

Furthermore, it is easy to see that $\left|G\right(\left[f\right]\left)\right|=2\dim \left(\right[f\left]\right)$. The proof of the next lemma is immediate; see [14] (IV.3, 3.18 Lemma).

Lemma 13.

Let $\kappa :(A,d)\to (A^{\prime },d^{\prime })$ be a map of metric spaces such that the following hold:

(i) 

κ is 1-Lipschitz.

(ii) 

κ is surjective.

(iii) 

$(A,d)$ is an antipodal metric space.

(iv) 

For each $x\in A$, there is $y\in A$ antipodal to x such that $d^{\prime }(\kappa \left(x\right),\kappa \left(y\right))=d(x,y)$.

• Then, it follows that κ is an isometry.

The next lemma follows immediately from Lemma 13; see [14] (IV.3, 3.19 Lemma).

Lemma 14.

Let $(X,d)$ be a totally split-decomposable metric space with an integer-valued metric satisfying the LRC. Let $\left[f\right]$ be any positive-dimensional cell of $\mathrm{E}(X,d)$ and let $\overline{f}$ and ${\psi }_{\overline{f}}$ be defined as in (iii) of Remark 2. Then, the map $\overline{\kappa }{:=\kappa |}_{\Gamma \left(\left[{\psi }_{\overline{f}}\right]\right)}:(\Gamma \left(\left[{\psi }_{\overline{f}}\right]\right),d_1)\to (G\left(\left[f\right]\right),d_{\infty })$ is an isometry.

For arbitrarily chosen $x,y\in X$, it follows from the definitions of ${\gamma }_{\left[{\psi }_{\overline{f}}\right]}^x$ and ${\gamma }_{\left[{\psi }_{\overline{f}}\right]}^y$ that we have

$$d_1({\gamma }_{[{\psi }_{\overline{f}}]}^x,{\gamma }_{[{\psi }_{\overline{f}}]}^y)=\sum _{S\in \mathcal{S}\left({\psi }_{\overline{f}}\right)}{\alpha }_S{\delta }_S(x,y),$$

(21)

where $\mathcal{S}\left({\psi }_{\overline{f}}\right)$ is weakly compatible. It follows by Theorem 5 that

$$(\Gamma \left(\left[{\psi }_{\overline{f}}\right]\right),d_1)\mathrm{is}\mathrm{a}\mathrm{totally}\mathrm{split}-\mathrm{decomposable}\mathrm{metric}\mathrm{space}.$$

(22)

4. The CAT(0) Link Condition for the Buneman Complex and the Cubical Injective Hull

The complex $B(\mathcal{S},\alpha )$ displays some similarities with the CAT(0) cube complex that is constructed in [19] and denoted by X. The next definition is a combinatorial characterization of the local CAT(0) condition for cube complexes (cf. [20]). We stress again that, unless otherwise stated, sets and complexes are in general infinite; in particular, $\mathcal{S}$ and X are in general infinite. Cell complexes are defined in Definition 2.

Definition 4.

A cell complex $\mathrm{K}$ with finite-dimensional cells that are combinatorial hypercubes is said to satisfy the CAT(0) link condition if, for every set of seven cells $C,C_1^1,C_1^2,C_1^3,C_2^1,C_2^2,C_2^3$ of $\mathrm{K}$ such that the following hold:

(i) 

$C={\bigcap }_{i\in \{1,2,3\}}{C}_2^i$,

(ii) 

$C_1^j={\bigcap }_{i\in \{1,2,3\}\setminus \left\{j\right\}}{C}_2^i$,

(iii) 

$\dim \left(C\right)=k\ge 0$, and for each $i\in \{1,2,3\}$ both $\dim \left(C_1^i\right)=k+1$ and $\dim \left(C_2^i\right)=k+2$,

• there exists a cell $\overline{C}$ of $\mathrm{K}$ such that $\dim \left(\overline{C}\right)=k+3$ and ${\bigcup }_{i\in \{1,2,3\}}{C}_2^i\subset \overline{C}$.

The next lemma follows directly from the definitions; see [14] (IV.4, 4.2 Lemma).

Lemma 15.

Let $(\mathcal{S},\alpha )$ be a split system pair on a set X. Then, the Buneman complex $B(\mathcal{S},\alpha )$ satisfies the CAT(0) link condition.

A split system $\mathcal{S}$ is called antipodal whenever, for every $x\in X$, there is $y\in X$ such that for every $S\in \mathcal{S}$ one has $S\left(x\right)\ne S\left(y\right)$. As a preliminary to the proof of Theorem 8, we require the following lemma (cf. [14], IV.4, 4.3 Lemma).

Lemma 16.

Let $\mathcal{S}$ be a split system on a set X. Then, the following hold:

(i) 

Assume that $\mathcal{S}$ is a weakly compatible split system and assume that the split system $\mathcal{S}\left({\mu }_2^i\right):=\mathcal{S}\left(\mu \right)\cup [\{S_1,S_2,S_3\}\setminus \left\{S_i\right\}]$ is antipodal for all $i\in \{1,2,3\}$. Then, $\mathcal{S}\left(\psi \right):=\mathcal{S}\left(\mu \right)\cup \{S_1,S_2,S_3\}$ is also antipodal.

(ii) 

Let $(X,d)$ be a totally split-decomposable metric space (hence, in particular, $\mathcal{S}$ is weakly compatible). Let ${\left\{f_2^i\right\}}_{i\in \{1,2,3\}}\subset {\mathrm{E}}^{\prime }(X,d)$ be such that, for the split systems provided in (i), one has $\mathcal{S}\left({\mu }_2^i\right)=\mathcal{S}\left({\psi }_{f_2^i}\right)$. Then, for every $x\in X$ it is possible to find $y\in X$ such that the following hold:

(a) 

For some $i\in \{1,2,3\}$, we have $\{x,y\}\in A\left(f_2^i\right)$.

(b) 

For every $S\in \mathcal{S}\left(\psi \right)$, we have $S\left(x\right)\ne S\left(y\right)$.

It is easy to prove (i) above by contradiction, and (ii) follows from (i) and the last statement of Lemma 8. Thus, we have now the tools at hand to prove the next theorem.

Theorem 8.

Let $(X,d)$ be a metric space with an integer-valued totally split-decomposable metric satisfying the LRC such that each cell of $\mathrm{E}(X,d)$ is a combinatorial hypercube. Then, $\mathrm{E}(X,d)$ satisfies the CAT(0) link condition.

Proof of Theorem 8.

Let $\left[f\right],\left[f_1^1\right],\left[f_1^2\right],\left[f_1^3\right],\left[f_2^1\right],\left[f_2^2\right],\left[f_2^3\right]\subset {\mathrm{E}}^{\prime }(X,d)=\mathrm{E}(X,d)$ be cells as in Definition 4 and with $f_j^i=\kappa \left({\psi }_{f_j^i}\right)$. Per Lemma 15, there is $\psi \in B(\mathcal{S},\alpha )$ such that $\mathrm{supp}\left(\psi \right)=\mathrm{supp}\left({\psi }_f\right)\cup {\bigcup }_{i\in \{1,2,3\}}{S}_i$ and $\mathcal{S}\left(\psi \right)=\mathcal{S}\left({\psi }_f\right)\cup \{S_1,S_2,S_3\}$ as in Lemma 16. Let $x\in X$ be chosen arbitrarily; per (ii) in Lemma 16, there is $y\in X$ such that for every $S\in \mathcal{S}\left(\psi \right)$ we have $S\left(x\right)\ne S\left(y\right)$, and without loss of generality $\{x,y\}\in A\left(f_2^1\right)$. By a direct computation, it can be shown that

$$\begin{array}{c}\hfill d_1({\varphi }_x,{\varphi }_y)=d_1({\varphi }_x,{\gamma }_{\left[\psi \right]}^x)+d_1({\gamma }_{\left[\psi \right]}^x,\psi )+d_1(\psi ,{\gamma }_{\left[\psi \right]}^y)+d_1({\gamma }_{\left[\psi \right]}^y,{\varphi }_y),\end{array}$$

(23)

similarly to (17). It follows that

$$\begin{array}{c}\hfill d(x,y)=d_1({\varphi }_x,\psi )+d_1(\psi ,{\varphi }_y)=\kappa \left(\psi \right)\left(x\right)+\kappa \left(\psi \right)\left(y\right).\end{array}$$

(24)

Because there is such an $y\in X$ for any $x\in X$, it follows that $\kappa \left(\psi \right)\in {\mathrm{E}}^{\prime }(X,d)$. Moreover, from the definition of $\psi$, we have

$$\bigcup _{i\in \{1,2,3\}}\kappa \left(\left[{\psi }_{f_2^i}\right]\right)=\kappa \left(\bigcup _{i\in \{1,2,3\}}\left[{\psi }_{f_2^i}\right]\right)\subset \kappa \left(\left[\psi \right]\right)\subset \left[\kappa \left(\psi \right)\right],$$

where the last inclusion follows from Lemma 10. Now, because $\left[\kappa \right(\psi \left)\right]$ is a hypercube, this proves that $\mathrm{E}(X,d)$ satisfies the CAT(0) link condition and concludes the proof. □

For any metric space $(X,d)$, the underlying graph $\mathrm{UG}(X,d)$ of $(X,d)$ is the graph $(X,E)$, where $\{x,y\}\in E$ if and only if $d(x,z)+d(z,y)>d(x,y)$ for any $z\in X\setminus \{x,y\}$. Furthermore, let $C_6$ denote the six-cycle metric graph and let $K_{3\times 2}$ denote the complete graph on six vertices with three disjoint edges taken away (i.e., the 1-skeleton of the octahedron).

Remark 3.

Note that if $\mathcal{S}$ is an antipodal split system on $(X,d)$, then for any ${\left(A_i\right)}_{i\in I}$, if ${\bigcup }_{i\in I}{A}_i=X$, it follows that ${\bigcap }_{i\in I}{A}_i=\varnothing$. Indeed, if $x\in {\bigcap }_{i\in I}{A}_i$, there is a subsystem of pairwise different splits ${\left\{S_i\right\}}_{i\in I}\subset \mathcal{S}$ such that $A_i=S_i\left(x\right)$. Now, we have $y\in X$ such that $y\in {\bigcap }_{S\in \mathcal{S}}\overline{S\left(x\right)}\subset {\bigcap }_{i\in I}\overline{S_i\left(x\right)}={\bigcap }_{i\in I}{A}_i^c={\left({\bigcup }_{i\in I}{A}_i\right)}^c$, which implies that ${\bigcup }_{i\in I}{A}_i\ne X$. An octahedral split system is an example of antipodal split system.

We can now proceed to prove the results described in the introduction.

Proof of Theorem 1.

The second part of Theorem 1, that is, the existence of $\mathrm{K}(X,d)$ and $\sigma$, follows immediately from Theorem 8. Indeed, Theorem 8 implies that if we re-metrize $\mathrm{E}(X,d)$ by identifying each cell (which is a parallelotope by the first part of Theorem 1) with a corresponding unit hypercube (of same dimension) endowed with the Euclidean metric, while considering the induced length metric, we obtain a complex $\mathrm{K}(X,d)$ which satisfies the CAT(0) link condition. Because $(X,d)$ satisfies the LRC, it follows that $\mathrm{K}(X,d)$ is complete and locally CAT(0) (analogous to I.7.13 Theorem and II.5.2 Theorem in [21]). Per the LRC, it also follows that $\mathrm{K}(X,d)$ is locally bi-Lipschitz equivalent to $\mathrm{E}(X,d)$; therefore, the topology induced by the length metric on $\mathrm{K}(X,d)$ is the same as the topology on $\mathrm{E}(X,d)$, meaning that $\mathrm{K}(X,d)$ is contractible as well. By the Cartan–Hadamard Theorem, it follows that $\mathrm{K}(X,d)$ is globally CAT(0).

We now prove the first part of Theorem 1. As an introductory remark, note that per Lemma 9, Lemma 14, and (22) it follows that $(G\left(\left[f\right]\right),d_{\infty })$ is an antipodal totally split-decomposable metric space with $2\dim \left(\right[f\left]\right)$ elements. By Theorem 7, $(\left[f\right],d_{\infty })$ is combinatorially equivalent to $\mathrm{E}(G\left(\left[f\right]\right),d_{\infty })$, which is per [6] (Theorem 1.2) an n-dimensional combinatorial hypercube if $\left|G\right(\left[f\right]\left)\right|=2n\ge 8$. Moreover, if $\left|G\right(\left[f\right]\left)\right|\le 4$, then $\mathrm{E}(G\left(\left[f\right]\right),d_{\infty })$ is clearly a combinatorial hypercube as well. Now, assume that $\left|G\right(\left[f\right]\left)\right|=6$. Because $(G\left(\left[f\right]\right),d_{\infty })$ is antipodal, it follows by [5] (Corollary 3.3) that $\mathrm{UG}(G\left(\left[f\right]\right),d_{\infty })$ is either $K_{3\times 2}$ or $C_6$. If $\mathrm{UG}(G\left(\left[f\right]\right),d_{\infty })=C_6$, then by [6] (Theorem 1.2 (a)), $\mathrm{E}(G\left(\left[f\right]\right),d_{\infty })$ is a three-dimensional combinatorial hypercube.

Assume now that $\mathrm{E}(G\left(\left[f\right]\right),d_{\infty })$ is a combinatorial rhombic dodecahedron, i.e., (ii) in Theorem 1 does not hold; then, $\mathrm{UG}(G\left(\left[f\right]\right),d_{\infty })=K_{3\times 2}$, and it follows by the proof of [5] (Theorem 5.1, Case 2) that $d_{\infty }(\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^x\right),\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^y\right))={\sum }_{S\in \{S_1,S_2,S_3,S_4\}}{\beta }_S{\delta }_S(x,y)$ where $\{S_1,S_2,S_3,S_4\}$ is weakly compatible and that the coefficients ${\beta }_S$ are all positive. Moreover, per (21) and Lemma 14, we have $d_{\infty }(\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^x\right),\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^y\right))={\sum }_{S\in \mathcal{S}\left({\psi }_{\overline{f}}\right)}{\alpha }_S{\delta }_S(x,y)$, where $\mathcal{S}\left({\psi }_{\overline{f}}\right)$ is weakly compatible and consists of d-splits of X. Note that the metric $d_{\infty }$ on $G\left(\right[f\left]\right)$ induces a pseudometric $\overline{d}$ on X by setting $\overline{d}(x,y):=d_{\infty }(\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^x\right),\kappa \left({\gamma }_{\left[{\psi }_{\overline{f}}\right]}^y\right))$. It follows by Theorem 5 and approximation by rescalings of integer-valued pseudometrics that $\mathcal{S}\left({\psi }_{\overline{f}}\right)=\{S_1,S_2,S_3,S_4\}=:\overline{\mathcal{S}}$, which has the form provided in (2).

From our introductory remark and since we have assumed that $\left[f\right]$ is a combinatorial rhombic dodecahedron, then $\left[f\right]$ must be a maximal cell (in dimensions higher than three, a cell must be a hypercube, and the same holds for all of its faces). Because our assumptions imply that $\mathrm{E}(X,d)={\mathrm{E}}^{\prime }(X,d)$ and $\left[f\right]$ is a three dimensional maximal cell, it follows by application of Zorn’s lemma that the graph $(X,A(f\left)\right)$ consists of three bipartite connected components. In fact, one can prove that for maximal cells those components are complete bipartite; this observation is due to Urs Lang (see [14], V.1, 1.2 Theorem). The respective partitions of the components are provided by $Y_1^1\bigsqcup Y_1^{-1}$, $Y_2^1\bigsqcup Y_2^{-1}$ and $Y_3^1\bigsqcup Y_3^{-1}$ (this is the only possibility, since if $\{x,y\}\in A\left(f\right)$, then $S\left(x\right)\ne S\left(y\right)$ for every $S\in \overline{\mathcal{S}}$). For each $S:=\{A,B\}\in \mathcal{S}\setminus \overline{\mathcal{S}}$, there is $\{x,y\}\in A\left(f\right)$ such that $S\left(x\right)=S\left(y\right)$ by bipartiteness (say, $\{x,y\}\subset A$). It follows from (10) that $\psi :={\psi }_{\overline{f}}\in \overline{T}(\mathcal{S},\alpha )$, where $\kappa \left(\psi \right)=\overline{f}$ and $\psi \left(A\right)=0$. Hence, $\psi \left(B\right)={\alpha }_S/2$, and thus for every further $\{x^{\prime },y^{\prime }\}\in A\left(f\right)$ one has $\{x^{\prime },y^{\prime }\}\not\subset B$. By bipartite completeness of $Y_i^1\bigsqcup Y_i^{-1}$, this implies that there are $\sigma ,\tau ,\theta \in \{\pm 1\}$ such that $Y_1^{\sigma }\cup Y_2^{\tau }\cup Y_3^{\theta }⊊A$, which is equivalent to $\{A,B\}$ and $\{A^{\prime },B^{\prime }\}=\{Y_1^{\sigma }\cup Y_2^{\tau }\cup Y_3^{\theta },Y_1^{-\sigma }\cup Y_2^{-\tau }\cup Y_3^{-\theta }\}\in \overline{\mathcal{S}}$ being compatible (i.e., $A^{\prime }\subset A$). It follows that (i) in Theorem 1 does not hold.

Conversely, assume that (i) does not hold, and hence there exists such a split subsystem $\overline{\mathcal{S}}$ with the properties stated in (i). Define $\psi \in H(\mathcal{S},\alpha )$ such that $\mathcal{S}\left(\psi \right)=\overline{\mathcal{S}}$ and $\overline{\mathcal{S}}$ consists of four splits, as provided in (2), and is a converse to (i). We can choose $\psi$ such that for any $S:=\{A,B\}\in \mathcal{S}\setminus \overline{\mathcal{S}}$ we additionally have $\psi \left(A\right)=0$ for $Y_1^{\sigma }\cup Y_2^{\tau }\cup Y_3^{\theta }⊊A$, and accordingly $\psi \left(B\right)={\alpha }_S/2$. By Remark 3, we have $f:=\kappa \left(\psi \right)\in \mathrm{E}(X,d)$. Thus, $(X,A(\kappa \left(\psi \right)\left)\right)$ consists of the three complete bipartite connected components $X={\cup }_{i\in \{1,2,3\}}(Y_i^1\bigsqcup Y_i^{-1})$ which implies that $\dim \left(\right[\kappa \left(\psi \right)\left]\right)=3$. It is easy to see that we have the decomposition $f={\sum }_{S\in \mathcal{S}}{\alpha }_S{f}_S$, so that for $S:=\{A,B\}\in \mathcal{S}$ we have

$$f_S\left(z\right)=\left\{\begin{array}{cc}\frac{\psi \left(A\right)}{{\alpha }_S/2}\hfill & \mathrm{if}z\in A,\hfill \\ \\ \frac{\psi \left(B\right)}{{\alpha }_S/2}\hfill & \mathrm{if}z\in B,\hfill \end{array}\right.$$

(25)

and ${\psi }_f=\psi$ holds for ${\psi }_f$ as defined in (iii) of Remark 2. From (i) of Remark 2, we have that ${\mathcal{S}}_f\subset {\mathcal{S}}_{\left[f\right]}$ and $|{\mathcal{S}}_{\left[f\right]}|=4$. However, (25) shows that $|{\mathcal{S}}_f|=4$, since $\left|\mathcal{S}\right(\psi \left)\right|=4$. It follows that ${\mathcal{S}}_f={\mathcal{S}}_{\left[f\right]}$, and we can thus set $\overline{f}:=f$. We then have ${\psi }_{\overline{f}}={\psi }_f=\psi$, and with Lemma 14 we obtain that $\left[f\right]$ is a combinatorial rhombic dodecahedron. Thus, (ii) in Theorem 1 does not hold either. This finishes the proof. □

We conclude now with the proof of another theorem.

Proof of Theorem 2.

Let $\Gamma$ be a finitely generated group with finite generating set G, and consider the associated word metric $d_G$ with respect to the alphabet $G\cup G^{-1}$. Recall that $I(x,y):=\{z\in \Gamma :d_G(x,z)+d_G(z,y)=d_G(x,y)\}$. Recall that for $x,v\in \Gamma$, the cone determined by the directed pair $(x,v)$ is provided by $C(x,v):=\{y\in \Gamma :v\in I(x,y\left)\right\}$ and that for a subset B of $\Gamma$, $\mathcal{C}\left(B\right)$ denotes the set of all pointed cones $(v,C(x,v\left)\right)$ with $v\in B$ and $x\in \Gamma$. Because $(\Gamma ,d_G)$ has $\beta$-stable intervals, [7] (Theorem 4.5 and Proposition 5.12) provide that $\mathrm{E}(\Gamma ,d_G)={\mathrm{E}}^{\prime }(\Gamma ,d_G)$ and that for any $z\in \Gamma$ and any $\alpha >0$, every $f\in {\mathrm{E}}^{\prime }(\Gamma ,d_G)$ with $f\left(z\right)\le \alpha$ satisfies $rk\left(A\left(f\right)\right)\le {\displaystyle \frac{1}{2}}\left|\mathcal{C}\left(B\right)\right|$. Therefore, $(\Gamma ,d_G)$ satisfies the local rank condition. It is implied by ([7], Theorem 1.1) that $\mathrm{E}(\Gamma ,d_G)$ is proper and has the structure of a polyhedral complex. The isometric action of $\Gamma$ on $(\Gamma ,d_G)$ provided by $(x,y)\mapsto L_x\left(y\right):=xy$ consequently induces a proper action by cell isometries of $\Gamma$ on $\mathrm{E}(\Gamma ,d_G)$, provided by $(x,f)\mapsto {\overline{L}}_x\left(f\right)=f\circ L_x^{-1}$ as a consequence of ([7], Theorem 1.4). Because $d_G$ is assumed to be totally split-decomposable and satisfies the combinatorial condition (i) in Theorem 1, it follows that the injective hull $E(\Gamma ,d_G)$ is combinatorially equivalent to a CAT(0) cube complex $\mathrm{K}(\Gamma ,d_G)$ via a canonical cell-wise affine isomorphism $\sigma :\mathrm{E}(\Gamma ,d_G)\to K(\Gamma ,d_G)$. This concludes the proof of (i).

Now, regarding (ii), if $(\Gamma ,d_G)$ is $\delta$-hyperbolic (in particular, if it has $\beta$-stable intervals), then $\mathrm{E}(\Gamma ,d_G)$ has only finitely many isometry types of cells, and the action is co-compact by virtue of ([7], Theorem 1.4). □

5. Examples

We conclude with two examples to illustrate our results and concepts.

Example 1.

Let $(X,d)$ denote an infinite connected bipartite $(4,4)$-graph as defined in [11], endowed with the shortest-path metric. From the finite case, it readily follows that d is totally split-decomposable, that is, $d={\sum }_{S\in \mathcal{S}}{\alpha }_S$, where every S is an alternating split on X (cf. [11]). Furthermore, $(X,d)$ satisfies the LRC. This can be seen by noting that the isometric cycles in $(X,d)$ are gated (cf. [11], Theorem 8.7), it follows that $(X,d)$ has 1-stable intervals; thus, by the proof of ([7], Theorem 1.1), we obtain the desired result. It is easy to see that by bipartiteness, the set of d-splits is octahedral-free. Hence, $(X,d)$ satisfies all assumptions of Theorem 1. Examples of such infinite bipartite $(4,4)$-graphs are provided for $m\ge 4$, for σ any element of the symmetric group ${\mathfrak{S}}_m$, and for ${\left\{r_{\sigma \left(i\right)\sigma (i+1)}\right\}}_{i\in \{1,\dots ,m\}}\subset (\mathbb{N}\cap [2,\infty ))\cup \{\infty \}$ by the Cayley graph of Coxeter groups of the form

$$\begin{array}{cc}\hfill C=〈s_1,\dots ,s_m|& {\left(s_{\sigma \left(1\right)}{s}_{\sigma \left(2\right)}\right)}^{r_{\sigma \left(1\right)\sigma \left(2\right)}}=1,\dots ,{\left(s_{\sigma (m-1)}{s}_{\sigma \left(m\right)}\right)}^{r_{\sigma (m-1)\sigma \left(m\right)}}=1,\hfill \\ \hfill & {\left(s_{\sigma \left(m\right)}{s}_{\sigma \left(1\right)}\right)}^{r_{\sigma \left(m\right)\sigma \left(1\right)}}=1.\hfill \end{array}$$

The restriction on the number of relations ensures that the Cayley graph is planar, $m\ge 4$ ensures that the degree is at least four, and the condition $r_{\sigma \left(i\right)\sigma (i+1)}\ge 2$ for every i ensures that each face contains at least four vertices.

Before moving on to the last and main example, we note that two splits $S:=\{A,B\}$ and $S^{\prime }=\{A^{\prime },B^{\prime }\}$ of X are called incompatible if $A\cap A^{\prime }, A\cap B^{\prime }, B\cap A^{\prime }, B\cap B^{\prime }\ne \varnothing$. A split system $\mathcal{S}$ is then called incompatible if any pair of splits in $\mathcal{S}$ is incompatible.

Example 2.

For $n\in \mathbb{N}$, let

$$C_{2n+1}:=\left(\{x_1,\dots ,x_{2n+1}\},{\left\{\{x_i,x_{i+1}\}\right\}}_{i\in \{1,\dots ,2n+1\}}\right)\mathrm{where}{x}_{2n+2}:=x_1.$$

The graph $C_{2n+1}$ is the odd cycle with $2n+1$ vertices, and we endow it with the shortest path metric d. We now provide an explicit description of the injective hull $\mathrm{E}(C_{2n+1},d)$. It can be easily verified that $d=\frac{1}{2}{\sum }_{S\in \mathcal{S}}{\delta }_S$, where $\mathcal{S}$ is the set of all d-splits of X; hence, $\alpha :\mathcal{S}\to (0,\infty )$ is constantly equal to $\frac{1}{2}$. We have $\mathcal{S}=\{S_1,\dots ,S_{2n+1}\}$, where the split $S_i$ is provided by $\{A_i,B_i\}=\{\{x_{i+1},\dots ,x_{i+n}\},\{x_{i+n+1},\dots ,x_i\}\}$ for $i\in \{1,\dots ,2n+1\}$, with the indices taken modulo $2n+1$. Now, it is not difficult to prove that the assumptions of Theorem 1 are fulfilled for $(X,d)=(C_{2n+1},d)$. Because we are in the case of a finite metric space, the LRC is trivially satisfied. Moreover, d is totally split-decomposable, and it is not difficult to see that $\mathcal{S}$ is octahedral-free. Thus, (i) in Theorem 1 holds.

Because $(C_{2n+1},d)$ is finite, we have $B(\mathcal{S},\alpha )=\overline{T}(\mathcal{S},\alpha )$ (cf. [22], Proposition 7.3), and by (iii) in Remark 2, the map $\kappa :\overline{T}(\mathcal{S},\alpha )\to \mathrm{E}(C_{2n+1},d)$ is in particular an isomorphism of cell complexes. Thus, the family of maximal cells of $\mathrm{E}(C_{2n+1},d)$ is in bijection with the family of maximal cells of $B(\mathcal{S},\alpha )$, which in turn bijectively corresponds with the family $\mathfrak{M}$ of maximal incompatible split subsystems $\mathcal{M}\subset \mathcal{S}$. To compute $\left|\mathfrak{M}\right|$, it is easier to describe the split system $\mathcal{S}$ in a different way. For $i\in \{1,\dots ,n+1\}$, we assign to every edge $\{x_i,x_{i+1}\}$ of $C_{2n+1}$ the pair of splits $S_i^1=\{A_i^1,B_i^1\}$ and $S_i^{-1}=\{A_i^{-1},B_i^{-1}\}$ which cut the edge $\{x_i,x_{i+1}\}$ (i.e., $S_i^1\left(x_i\right)\ne S_i^1\left(x_{i+1}\right)$ and $S_i^{-1}\left(x_i\right)\ne S_i^{-1}\left(x_{i+1}\right)$) and which are determined by the requirements $x_i\in A_i^1$ and $|A_i^1|=n+1$ along with $x_i\in A_i^{-1}$ and $|A_i^{-1}|=n$; moreover, $S_1^1=S_{n+1}^{-1}$. We divide the family $\mathcal{M}$ of maximal incompatible split subsystems of $\mathcal{S}$ into three subfamilies

$$\mathfrak{M}={\mathfrak{M}}^a\cup {\mathfrak{M}}^b\cup {\mathfrak{M}}^c,$$

so that for $\mathcal{M}:=\{S_{j_1}^{{\sigma }_1},\dots ,S_{j_k}^{{\sigma }_k}\}\in \mathfrak{M}$ with $1\le j_1<j_2<\dots <j_{k-1}<j_k\le 2n+1$ we have three cases for each of the three subsets of $\mathfrak{M}$.

First, $\mathcal{M}\in {\mathfrak{M}}^a$ if and only if $S_{j_1}^{{\sigma }_1}=S_1^1$. In this case, we have

$$|{\mathfrak{M}}^a|={\mathrm{\Sigma }}_{(n-1,n-1)}^a+{\mathrm{\Sigma }}_{(n-1,n)}^a,$$

where ${\mathrm{\Sigma }}_{(n-1,n-1)}:={\mathrm{\Sigma }}_{(n-1,n-1)}^a$ stands for the case in which $S_{j_k}^{{\sigma }_k}=S_n^1$ and ${\mathrm{\Sigma }}_{(n-1,n)}:={\mathrm{\Sigma }}_{(n-1,n)}^a$ for the case in which $S_{j_k}^{{\sigma }_k}=S_n^{-1}$. The notation ${\mathrm{\Sigma }}_{(n-1,n-1)}$ refers to the fact that $|B_{j_1}^{{\sigma }_1}\cap A_{j_k}^{{\sigma }_k}|=n-1$ and $|A_{j_1}^{{\sigma }_1}\cap B_{j_k}^{{\sigma }_k}|=n-1$ (i.e., starting with $x_2$ and going counterclockwise, we count $n-1$ points until we hit the line representing $S_{j_k}^{{\sigma }_k}$; similarly, when starting with $x_{n+2}$, we count $n-1$ points until we hit the line representing $S_{j_k}^{{\sigma }_k}$). This case is depicted in Figure 1.

Second, $\mathcal{M}\in {\mathfrak{M}}^b$ if and only if $S_{j_1}^{{\sigma }_1}=S_1^{-1}$. In this case, we have

$$|{\mathfrak{M}}^b|={\mathrm{\Sigma }}_{(n-1,n-1)}^b+{\mathrm{\Sigma }}_{(n,n-1)}^b,$$

where by symmetry ${\mathrm{\Sigma }}_{(n-1,n-1)}^b={\mathrm{\Sigma }}_{(n-1,n-1)}$ stands for the case in which $S_{j_k}^{{\sigma }_k}=S_n^{-1}$ and ${\mathrm{\Sigma }}_{(n,n-1)}^b={\mathrm{\Sigma }}_{(n-1,n)}$ for the case in which $S_{j_k}^{{\sigma }_k}=S_{n+1}^1$.

Third, $\mathcal{M}\in {\mathfrak{M}}^c$ if and only if $S_{j_1}^{{\sigma }_1}=S_2^1$. In this case, we have

$$|{\mathfrak{M}}^c|={\mathrm{\Sigma }}_{(n-1,n-1)}^c,$$

and by symmetry ${\mathrm{\Sigma }}_{(n-1,n-1)}^c={\mathrm{\Sigma }}_{(n-1,n-1)}$ stands for the case in which $S_{j_k}^{{\sigma }_k}=S_{n+1}^1$.

Summing up, we obtain the formula

$$\left|\mathfrak{M}\right|={\mathrm{\Theta }}_n=3{\mathrm{\Sigma }}_{(n-1,n-1)}+2{\mathrm{\Sigma }}_{(n-1,n)},$$

and it is easy to see that we also have the following recurrence relations ${\mathrm{\Sigma }}_{(n-1,n-1)}={\mathrm{\Sigma }}_{(n-2,n-2)}+{\mathrm{\Sigma }}_{(n-3,n-2)}$ and ${\mathrm{\Sigma }}_{(n-1,n)}={\mathrm{\Sigma }}_{(n-2,n-1)}+{\mathrm{\Sigma }}_{(n-2,n-2)}$ along with the initial conditions ${\mathrm{\Sigma }}_{(0,0)}:=1$, ${\mathrm{\Sigma }}_{(0,1)}:=0$, and ${\mathrm{\Sigma }}_{(1,1)}:=1$. The first values of ${\mathrm{\Theta }}_n$ for $n\ge 1$ are listed in the following table.

n	1	2	3	4	5	6	7	8	9	10
$C_{2n+1}$	$C_3$	$C_5$	$C_7$	$C_9$	$C_{11}$	$C_{13}$	$C_{15}$	$C_{17}$	$C_{19}$	$C_{21}$
${\mathrm{\Theta }}_n$	3	5	7	12	22	39	68	119	209	367

The numbers ${\mathrm{\Theta }}_n$ can also be proved (see e.g., [23]) to correspond to the coefficient of $z^n$ in the power series expansion in a neighborhood of the origin of the analytic function

$$z\mapsto {\displaystyle \frac{3z-z^2}{1-2z+z^2-z^3}}.$$