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)
of a set
X is a pair of non-empty subsets of
X such that
and
, or in other words,
. For
, we denote by
the element of
S that contains
x. The
split (pseudo-)metric associated with
S is then a pseudometric
on
X such that
For a pseudometric
d on
X, we refer to
as a
d-split (of
X) if the
isolation index provided by
satisfies
. The pseudometric
d is called
totally split-decomposable if
, where
is the set of all
d-splits. A split subsystem
is called
octahedral if and only if there is a partition of
X into a disjoint union of six non-empty sets
such that
consists of the following four splits:
while
is called
octahedral-free if it does not contain any octahedral split subsystem. Two splits
and
are said to be
compatible if
(and thus
), or alternatively if
(and thus
).
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
, let us consider the vector space
of real-valued functions on
X and
. We refer to
as
extremal if there is no
in
distinct from
f. The set
of extremal functions is equivalently provided by
In order to be able to describe the structure of
further, it is possible to assign the undirected graph with vertex set
X and edge set
to every
, allowing self-loops
which correspond to zeros of
f. Furthermore, we let
. Note that if
, then the graph
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
with
, and we denote by
the collection of admissible sets. To every
, we associate the affine subspace
of
provided by
. We define the
rank of
A by
, which is provided by the number of bipartite components or
even A-components of
. As an example, if
denotes the distance function to
i.e.,
, then
. We also have another example to show that the rank can be infinite; indeed, if
denotes the unit circle endowed with the intrinsic geodesic metric and if
denotes the function that is constantly equal to
, then it is possible to verify that
. See [
9] for a discussion of related examples.
If
is a finite metric space, then
is a finite polyhedral complex; if
is infinite, then we say that
satisfies the
local rank condition (LRC) if and only if for every
there exist
such that for all
with
we have
. Recall (cf. [
7], Theorem 4.5) that if
is a metric space with an integer-valued metric and satisfying the (LRC), then
; in this case, let
. Then, the family
defines a polyhedral structure on
; in particular,
is a face of
if and only if
. This defines a canonical locally finite dimensional polyhedral structure on
. In the case where
d is totally split-decomposable, our goal is to provide necessary and sufficient conditions ensuring that
is combinatorially equivalent to a
cube complex. Accordingly, we have the following theorem.
Theorem 1. Let be a metric space with an integer-valued totally split-decomposable metric satisfying the local rank condition and let be the set of all d-splits; then, the following are equivalent:
- (i)
does not contain any octahedral split subsystem , satisfying the requirement that for every there is such that S and are compatible.
- (ii)
Each cell of is a parallelotope.
If (i) or equivalently (ii) in Theorem 1 holds, then there is a CAT(0) cube complex and a canonical bijective cell complex isomorphism 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
. The difference between two elements of
has a finite
-norm, and
is endowed with the metric
. It is easy to see that for
, if
, then
; hence, if
is a pseudometric space and
is the associated metric space obtained by collapsing every maximal set of diameter zero to a single point, then
and
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
are uniformly bounded, then
in Theorem 1 can be chosen to be bi-Lipschitz. For a metric space
, let
denote the
interval between
x and
y, and for
let the
cone determined by the directed pair
be provided by
. For a subset
B of
X, we denote by
the set of all pointed cones
with
and
. In addition,
is called
discretely geodesic if the metric is integer-valued and for every pair of points
there exists an isometric embedding
such that
and
. Moreover, we say that a discretely geodesic metric space
X has
β-stable intervals for some constant
if, for every triple of points
such that
, we have
, where
denotes the Hausdorff distance in
X.
Among its other features, the injective hull has applications to geometric group theory. Let be a finitely generated group and G a finite generating set, and let be equipped with the word metric with respect to the alphabet . Note that the metric is integer-valued.
Theorem 2. Let be a metric space where Γ is a finitely generated group and G a finite generating set, and let denote the word metric defined on Γ with respect to the alphabet . Assume that is totally split-decomposable and that (i) or equivalently (ii) in Theorem 1 holds. Then, the following hold:
- (i)
If has β-stable intervals, then there is a proper action of Γ (a CAT(0) cube complex ) provided by - (ii)
If is δ-hyperbolic, then the action of Γ on 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
. The
isolation index of a pair
of non-empty subsets with respect to a pseudometric
d on
X is the non-negative number
(equivalently
, or simply
) provided by (
1). We then refer to a pseudometric
on
X as
split-prime if
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 be a pseudometric space with an integer-valued pseudometric, let be the set of all splits of X, and let be the set all d-splits. Letting is a pseudometric such that for every split we have ; in particular, there is a split-prime pseudometric such that 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 be a metric space with an integer-valued metric satisfying the LRC. Let be the family of all d-splits of X such that , and let for every . Then, setting , we haveMoreover, for any split of X and any such that , if , then the following hold: - (i)
S is a d-split of X.
- (ii)
.
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
as defined at the beginning of the introduction is a subset of
. Note that a function
belongs to
if and only if
where
denotes the distance function to
, i.e.,
. The metric
on
is thus well-defined and
is equipped with the induced metric. We have the canonical isometric mapping
, provided by
. Let
be any pseudometric space. A
partial split of
X is a pair of non-empty subsets of
X such that
. If
also holds, then
is a split of
X. A
partial d-split is a partial split
for which
. Note now that if
is a pseudometric space with integer-valued pseudometric, then for any split
S we have
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
there are
and
, such that
(if
, we still use
to denote
). Moreover, if
satisfies
, then we have
.
Theorem 4. Let be a pseudometric space such that, for every , there are only finitely many distinct d-splits S satisfying . Moreover, let be a partial d-split. Then, 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
.
Lemma 1. Let be a pseudometric space with an integer-valued pseudometric. For every , there are at most distinct d-splits S satisfying .
Proof. Assume by contradiction that we can find
distinct
d-splits
such that
. For every
, choose
such that either
or
. Furthermore, setting
, we have
if
; hence
and per the special case ([
1], Theorem 1) of Theorem 4 applied to the partial split
and
of the finite set
Z, the right-hand side is less than or equal to
, 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 be a pseudometric space with an integer-valued pseudometric. Let be the set of all d-splits of X and let be any finite subset. If for every and for every other split, then is a pseudometric such that for every split S of X we have 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 of splits of a set X as weakly compatible if there are no four points and three splits such that for any one has .
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 denote any pseudometric space with an integer-valued metric; then, the d-splits of X are weakly compatible. Conversely, let denote any collection of weakly compatible splits of X and choose for each some such thatIt follows that is the set of all d-splits and that for each , the isolation index coincides with . 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
is a polyhedral complex. We denote by
the set of vertices of the polyhedral complex
; equivalently, it is the set of all functions
such that
. 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 be a metric space with an integer-valued metric satisfying the LRC, and let . Let be any finite subset of the set of all d-splits of X. If we pick for every , then we have functions such that Proof. Let
, where
is any finite subset
. Note that
means that
f is a vertex of the polyhedral complex
, which is equivalent to
by definition. Thus,
is in particular not bipartite, meaning that there are
such that
and either
or
. Assume without loss of generality that
. Note also that if there are
such that we have
, then
, which contradicts our assumption. Hence, for any
, we have
. Now, setting
it readily follows that
and
satisfy
. Furthermore, note that if
, then either
or
and
. In both cases, it is evident that
(noting that
refers to those pairs
satisfying
). Thus,
hence, in particular,
. For
, we can proceed as above and find
such that
. Now, by Theorem 3, it follows that
. This implies that for any
, we have
. Now, setting
we obtain
, as before. Per (
7), it follows that
. It then follows as a result that
where
. Proceeding by induction, we get the desired result. □
For an infinite split system , we generally have and cannot replace by .
Theorem 6. Let be a metric space with an integer-valued metric satisfying the LRC. Let be the family of all d-splits of X such that and let for every , setting . Then, Proof. Let
be the set of all
d-splits and let
. As in the proof of Lemma 3,
means that
f is a vertex of the polyhedral complex
, which by definition is equivalent to
. Thus,
is not bipartite, which means that there are
such that
. Without loss of generality, we assume that
. Then, for any
, we have
. Now, setting
we first show that for every
, we have
. Note that for every
there exists
such that
. Furthermore, we have
Because
, we deduce that the set
is finite. Moreover, for every
, because
, by definition we have
and
. It then follows that
as well as that
For
, it now remains to show that
. For every
, there are
such that
. Because for every
we have
, it follows that
, where
. By Lemma 3 and setting
, it follows that
and thus that
. This shows that
. Now, using the fact that
satisfies the LRC, we can consider for each cell of
all finite convex combinations of its vertices, and by convexity of
and
, for every
we can deduce that
. Finally, adding
on both sides and intersecting with
, we get
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
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 such that one has for every . From (4), we obtain a decomposition of g as . Moreover, ; hence,which is the desired result. 3. The Buneman Complex and Related Topics
If is a split system on a set X and is any map , then the pair is called a split system pair (of X). If is weakly compatible, as in Definition 1, then is called a weakly compatible split system pair. Now, let be a weakly compatible split system on a pseudometric space with an integer-valued pseudometric and assume that . Per Theorem 5, is the set of all d-splits of X; thus, d totally split-decomposable. The weakly compatible split system pair is called the split system pair associated with . Unless otherwise stated, this is the split system pair that we refer to in the rest of the discussion when considering a totally split-decomposable pseudometric space . We want to stress that the sets and X are generally infinite.
Definition 2. We refer to as acell complexif K is a subset of a real vector space endowed with a family of convex subsets of such that the collection verifies that and for any . The sets are called the cells of , and the dimension of is the dimension of its affine hull, which in general is infinite.
Let
be any split system pair on a set
X, and consider
For a map
, we can write
. If
, then we denote the complement
by
. For a given
, we denote the associated split
by
. We define the following hypercube:
which is in general infinitely dimensional. Here,
has a natural cell complex structure; cells are sets of the form
where
. The cells of
are (possibly) infinite-dimensional hypercubes. The
Buneman complex is the subcomplex of
provided by
Next, we define
It is easy to see that
is a subcomplex of
, as in the finite case ([
16],
Section 4). For
, the map
is defined as
Furthermore, let
be provided by
The map
is provided by
, where
for
. As a direct application of the above definitions, we obtain the following lemma.
Lemma 4. Let be a split system pair on a set X and assume that defines a pseudometric on X. Then, the following hold:
- (i)
For every , we have .
- (ii)
For every , we have .
- (iii)
, where each side might be infinite.
Under the assumptions of Lemma 4, for
and
, let
if
and
if
. For a further
and
, a direct calculation shows that
and that equality holds if
.
Lemma 5. Let be a split system pair on a set X and assume that defines a pseudometric on X. Then, for every , the following are equivalent:
- (i)
.
- (ii)
.
Proof. Consider
. We first show that
implies
. Let
, and assume that
. For
, we have
; thus,
. By our contradiction assumption, there is
such that for every
we have
, where the left-hand side is possibly infinite. Because
, equality holds in (
10); hence, there is
such that
and
. Therefore,
where
. Moreover,
. It follows that
. This shows that
implies
.
To show the other implication, assume that
. For every
, there is
such that
. Per (
10), we have the following for any
:
Note that for any
, there exists by definition
as well as
such that
. Now, if
and assuming by contradiction that
, we can pick an arbitrary
. Then, for every
we have
, and thus
. It is now clear that the existence of
z contradicts (
11). Indeed, for any
, there is
such that
, and hence
However,
and
y can be chosen so that
. Thus, by (
11) and (
12), we have
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
.
Lemma 6. Let be a totally split-decomposable metric space with an integer-valued metric satisfying the LRC. Then, is surjective.
For a map
, let
. Define for a cell
of
and
the map
provided by
Note that if
and
, then
. Therefore,
; hence,
by (
9).
Definition 3. Let be an isometric map of pseudometric spaces. We say that is X-gated (for i and with respect to ) if and only if for every there is such that for every we have .
From Lemma 5, it follows that the restriction of
to
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 be a split system pair on a set X and assume that defines a pseudometric on X. Then, every cell of is X-gated with respect to the restriction of to .
By virtue of Lemma 5, we also obtain the next lemma.
Lemma 8. Let be a split system pair on a set X and assume that defines a pseudometric on X. Then, for every , the split system is antipodal, which means that for any there is such thatFor , if , then x and y satisfy (14). Proof. Let
. By Lemma 5, we have
. Thus, for any
, there is
such that
, which can be rewritten as
It is easy to see that for every
, we have
which together with (
15) implies
Assume now that there is
such that
; then, per (
16) we have
, which implies
and thus
, which is a contradiction. This finishes the proof. □
By Lemma 7, every cell
of
is
X-gated. Let
denote the set of all
X-gates of
endowed with the restriction of
. A pseudometric space
is called
antipodal if there exists an involution
such that for every
one has
. 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 be a split system pair on a set X and assume that defines a pseudometric on X. Then, for every cell of , the metric space is antipodal.
For , we know from Lemma 5 that . Recalling from the introduction that and setting , we let . If has an integer-valued metric and satisfies the LRC, is a cell complex in which all cells are of this form.
Lemma 10. Let be a split system pair on a set X and assume that defines a pseudometric on X. Then, for every cell of , one has .
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
such that
we have the following equality for every
:
which implies
.
Remark 2. For , let , where . Set and , then:
- (i)
Assume that 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 of can be written as where for each one has . To see that this holds, note first that per Theorem 6; then, it is easy to see from the definition of the sets and the fact that we have a decomposition with for every that where and . In addition, for every , we have for each satisfying .
- (ii)
It is not difficult to see that if is as in (i) and if every cell of is a combinatorial hypercube, then the representation in (18) verifies 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 is as usual and if , then it is easy to see that ; in particular, .
- (iii)
For , as in (ii), let us define by the assignment , where is defined for every as well as for arbitrarily chosen and by setting This definition depends on a choice of a representation for f, and in general this choice is not unique. We denote by an arbitrarily chosen element of such that and that is maximal among the elements of . Furthermore, note that we always have . It follows that κ is surjective. In general, ; however, if every cell of is a combinatorial hypercube, then the map 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 be a totally split-decomposable metric space with an integer-valued metric which satisfies the LRC. For every cell of , if and are defined as in (iii) of Remark 2, then one has .
Recall that there is a canonical isometric embedding
provided by
in which
is endowed with the metric
. Assume that
satisfies the assumptions of Lemma 11. We say that
is
cell-decomposable if every cell
C of
is
X-gated (cf. Definition 3). Now, a direct computation shows that
is a gate for
in
. Thus, we have the following lemma; see also [
14] (IV.3, 3.16 Lemma).
Lemma 12. Let be a totally split-decomposable metric space with an integer-valued metric satisfying the LRC. Then, is cell-decomposable.
Let
denote the set of all
X-gates of
endowed with the restriction of the metric
. We denote by
the gate of
in
. It is easy to see that the proof of ([
6], Theorem 1.1) directly generalizes to the case where
as long as
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 be a metric space with an integer-valued metric satisfying the LRC. If is such that is X-gated, then the following hold:
- (i)
is a finite antipodal metric space.
- (ii)
The map provided by is a bijective isometry as well as an isomorphism of polytopes.
Furthermore, it is easy to see that
. The proof of the next lemma is immediate; see [
14] (IV.3, 3.18 Lemma).
Lemma 13. Let be a map of metric spaces such that the following hold:
- (i)
κ is 1-Lipschitz.
- (ii)
κ is surjective.
- (iii)
is an antipodal metric space.
- (iv)
For each , there is antipodal to x such that .
The next lemma follows immediately from Lemma 13; see [
14] (IV.3, 3.19 Lemma).
Lemma 14. Let be a totally split-decomposable metric space with an integer-valued metric satisfying the LRC. Let be any positive-dimensional cell of and let and be defined as in (iii) of Remark 2. Then, the map is an isometry.
For arbitrarily chosen
, it follows from the definitions of
and
that we have
where
is weakly compatible. It follows by Theorem 5 that
4. The CAT(0) Link Condition for the Buneman Complex and the Cubical Injective Hull
The complex
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,
and
X are in general infinite. Cell complexes are defined in Definition 2.
Definition 4. A cell complex with finite-dimensional cells that are combinatorial hypercubes is said to satisfy the CAT(0) link condition if, for every set of seven cells of such that the following hold:
- (i)
,
- (ii)
,
- (iii)
, and for each both and ,
The next lemma follows directly from the definitions; see [
14] (IV.4, 4.2 Lemma).
Lemma 15. Let be a split system pair on a set X. Then, the Buneman complex satisfies the CAT(0) link condition.
A split system
is called antipodal whenever, for every
, there is
such that for every
one has
. As a preliminary to the proof of Theorem 8, we require the following lemma (cf. [
14], IV.4, 4.3 Lemma).
Lemma 16. Let be a split system on a set X. Then, the following hold:
- (i)
Assume that is a weakly compatible split system and assume that the split system is antipodal for all . Then, is also antipodal.
- (ii)
Let be a totally split-decomposable metric space (hence, in particular, is weakly compatible). Let be such that, for the split systems provided in (i), one has . Then, for every it is possible to find such that the following hold:
- (a)
For some , we have .
- (b)
For every , we have .
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 be a metric space with an integer-valued totally split-decomposable metric satisfying the LRC such that each cell of is a combinatorial hypercube. Then, satisfies the CAT(0) link condition.
Proof of Theorem 8. Let
be cells as in Definition 4 and with
. Per Lemma 15, there is
such that
and
as in Lemma 16. Let
be chosen arbitrarily; per (ii) in Lemma 16, there is
such that for every
we have
, and without loss of generality
. By a direct computation, it can be shown that
similarly to (
17). It follows that
Because there is such an
for any
, it follows that
. Moreover, from the definition of
, we have
where the last inclusion follows from Lemma 10. Now, because
is a hypercube, this proves that
satisfies the CAT(0) link condition and concludes the proof. □
For any metric space , the underlying graph of is the graph , where if and only if for any . Furthermore, let denote the six-cycle metric graph and let 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 is an antipodal split system on , then for any , if , it follows that . Indeed, if , there is a subsystem of pairwise different splits such that . Now, we have such that , which implies that . 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
and
, follows immediately from Theorem 8. Indeed, Theorem 8 implies that if we re-metrize
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
which satisfies the CAT(0) link condition. Because
satisfies the LRC, it follows that
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
is locally bi-Lipschitz equivalent to
; therefore, the topology induced by the length metric on
is the same as the topology on
, meaning that
is contractible as well. By the Cartan–Hadamard Theorem, it follows that
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
is an antipodal totally split-decomposable metric space with
elements. By Theorem 7,
is combinatorially equivalent to
, which is per [
6] (Theorem 1.2) an
n-dimensional combinatorial hypercube if
. Moreover, if
, then
is clearly a combinatorial hypercube as well. Now, assume that
. Because
is antipodal, it follows by [
5] (Corollary 3.3) that
is either
or
. If
, then by [
6] (Theorem 1.2 (a)),
is a three-dimensional combinatorial hypercube.
Assume now that
is a combinatorial rhombic dodecahedron, i.e., (ii) in Theorem 1 does not hold; then,
, and it follows by the proof of [
5] (Theorem 5.1, Case 2) that
where
is weakly compatible and that the coefficients
are all positive. Moreover, per (
21) and Lemma 14, we have
, where
is weakly compatible and consists of
d-splits of
X. Note that the metric
on
induces a pseudometric
on
X by setting
. It follows by Theorem 5 and approximation by rescalings of integer-valued pseudometrics that
, which has the form provided in (
2).
From our introductory remark and since we have assumed that
is a combinatorial rhombic dodecahedron, then
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
and
is a three dimensional maximal cell, it follows by application of Zorn’s lemma that the graph
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
,
and
(this is the only possibility, since if
, then
for every
). For each
, there is
such that
by bipartiteness (say,
). It follows from (
10) that
, where
and
. Hence,
, and thus for every further
one has
. By bipartite completeness of
, this implies that there are
such that
, which is equivalent to
and
being compatible (i.e.,
). 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
with the properties stated in (i). Define
such that
and
consists of four splits, as provided in (
2), and is a converse to (i). We can choose
such that for any
we additionally have
for
, and accordingly
. By Remark 3, we have
. Thus,
consists of the three complete bipartite connected components
which implies that
. It is easy to see that we have the decomposition
, so that for
we have
and
holds for
as defined in (iii) of Remark 2. From (i) of Remark 2, we have that
and
. However, (
25) shows that
, since
. It follows that
, and we can thus set
. We then have
, and with Lemma 14 we obtain that
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
be a finitely generated group with finite generating set
G, and consider the associated word metric
with respect to the alphabet
. Recall that
. Recall that for
, the cone determined by the directed pair
is provided by
and that for a subset
B of
,
denotes the set of all pointed cones
with
and
. Because
has
-stable intervals, [
7] (Theorem 4.5 and Proposition 5.12) provide that
and that for any
and any
, every
with
satisfies
. Therefore,
satisfies the local rank condition. It is implied by ([
7], Theorem 1.1) that
is proper and has the structure of a polyhedral complex. The isometric action of
on
provided by
consequently induces a proper action by cell isometries of
on
, provided by
as a consequence of ([
7], Theorem 1.4). Because
is assumed to be totally split-decomposable and satisfies the combinatorial condition (i) in Theorem 1, it follows that the injective hull
is combinatorially equivalent to a CAT(0) cube complex
via a canonical cell-wise affine isomorphism
. This concludes the proof of (i).
Now, regarding (ii), if
is
-hyperbolic (in particular, if it has
-stable intervals), then
has only finitely many isometry types of cells, and the action is co-compact by virtue of ([
7], Theorem 1.4). □