Geometric algebra for sets with betweenness relations

Given a betweenness relation on a nonempty set E, a certain abelian group T=TE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}= {{\mathbb {T}}}_E$$\end{document} given in terms of generators and relations is investigated. This group controls the given betweenness relation in an algebraic form. That is, the group structure algebraically unfolds geometric relations, and in turn allows us to read off geometric properties from algebraic relations emerging from them. The most important examples for betweenness relations arise from ordered sets on the one side and from intervals in metric spaces on the other side. The structure of T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {T}}$$\end{document} will be determined completely in case of totally ordered sets as well as for several classes of metric spaces.


Introduction
There are several motivations for this paper.
A concrete one is the so-called Tutte group of a matroid, which was introduced by Andreas Dress and the second named author in Dress and Wenzel (1989)  Universität Leipzig, Mathematisches Institut, Augustusplatz 10, 04109 Leipzig, Germany geometric algebra for matroids. This abelian group is defined in terms of generators and relations, and it reflects properties of determinants and hyperplane functions.
From a general perspective, while a metric expresses pairwise relations, for a deeper understanding the geometry of a metric space, one should also look at relations between more than two points. In particular, as the triangle inequality involves three points, one can systematically analyze to what extent triangle inequalities fail to be equalities. This leads to the notion of Gromov products Gromov (1980), and more generally, to abstract curvature concepts for metric spaces, see for instance Joharinad and Jost (2019). When the triangle inequality is an exact equality, one point lies between the other two. This can be axiomatized in terms of a ternary relation, the betweenness relation. This can again be formulated in abstract terms, and it connects metric and pre-or partially ordered spaces (in such a space, b is between a and c, if a < b < c or c < b < a). But there are also metric spaces where an even stronger relation occurs. The spaces we have in mind here are the tripod spaces where for any three points, there is a fourth point (not necessarily distinct from those three), a median that sits between any two of them. It is the purpose of this paper to develop an algebraic framework that can encode such relationships in a flexible manner. This is the Dress group that we shall introduce below, which captures betweenness, tripod and other such relations systematically and translates them into the algebraic structure of a group.
A betweenness relation on a nonempty set E is a ternary relation * defined on E satisfying certain axioms. For a general overview about betweenness relations, see in particular Adeleke and Neumann (1998). The most important examples are induced by ordered sets or by metric spaces. Here, we are mainly interested in metric spaces M = (E, d), where A * B * C for pairwise distinct elements A, B, C ∈ E means that d(A, C) = d(A, B) + d (B, C).
We want to point out that, in particular within an axiomatic approach to Euclidean Geometry, ordered sets and metrics are closely related: Any Euclidean line may be provided with two inverse complete orderings in a canonical way. As, in general, none of these two orderings may be preferred over the other one, it seems more natural to study the common betweenness relation. This is exactly the same betweenness relation as given by the Euclidean distance. See in particular Böhm et al. (1988) and Borsuk and Szmielew (2018).
For any betweenness relation defined on a nonempty set E, we define the Dress group, an abelian group T E that is generated by the elements T A,B for A, B ∈ E with A = B and satisfies the following relations: Remark 1. 1 We name this group after our friend and mentor Andreas Dress, because he pioneered the interaction of geometric and algebraic concepts, and the group T E is constructed in that spirit.
Remark 1. 2 We point out that we do not define elements T A,A , although one might think that one should stipulate that T A,A = 0 in the group. This is consistent with the fact that in the Definition 2.4 of a betweenness relation it is required that a point that is between two others is different from both of them.
If the betweenness relation is defined in terms of a metric d, this means in particular-for pairwise distinct elements A, B, C:

T A,C = T A,B + T B,C whenever d(A, C) = d(A, B) + d(B, C).
As Corollary 3.7 shows, the converse is also true. Hence, the Dress group T E controls whether three pairwise distinct elements A, B, C fulfill equality in the triangle inequality.
In particular, it turns out that for two given metrics defined on the same set E, the corresponding groups coincide, if both of these metrics give rise to exactly the same intervals in E. More precisely, see Proposition 3.10. Particularly, the Dress groups of two homoeomorphic trees are isomorphic. In other words: Topological deformations of a tree do not change the Dress group, because the intervals in the tree remain the same.
However, in general two homoeomorphic metric spaces may exhibit quite different Dress groups, because both spaces might have quite different intervals.
In more abstract terms, we let pairs of distinct points in a set generate a group. In a metric space, or more generally, in a space with a betweenness relation, betweenness is translated into a relation in the group. Since the group structure unfolds the relations between the generators, we can then read off algebraic consequences of geometric relations and use them to infer geometric properties. A first instance is Theorem 5.8: The Dress group of a metric space is torsion free if for any two distinct points there exists a unique geodesic trace between these points. In other words: If the Dress group is not torsion free, there must exist two points with two distinct geodesic traces between these points. Thus, a simple algebraic property of the group tells us something about the underlying geometric structure.
However, the converse of Theorem 5.8 is not true, see Example 5.12. In fact, when there is more than one shortest geodesic between two points, a nontrivial relation in the Dress group emerges. In that example, there is only such pair of points, and so, there is only a single relation, and that does not yet create a torsion element in the group.
In this paper, we start exploring the Dress group. For certain geometric spaces and for certain graphs, in particular for cycles, we shall completely determine the structure of T E .
A complete determination will also be achieved for the plane R 2 with the Manhattan Metric d 1 -as well as for R-trees and discrete trees.
This also provides a link to another motivation for this paper which at first sight is quite different. This comes from abstract metric geometry. In fact, in many branches of mathematics, geodesics in metric spaces (or, more precisely, their traces) are considered to be topologically connected sets, in particular within Minkowski geometries or in differential geometry. But, on the other hand, shortest paths in discrete graphs are also called geodesics. In particular, these geodesics are important within the theory of buildings, but, within their geometric realizations, geodesics that are homeomorphic to line segments, are important as well; see Brown (1989).
In this paper, we present a unified approach to geodesics in arbitrary metric spaces M = (E, d) with at least two elements: For distinct x, y ∈ E, a shortest geodesic from x to y is a map g : J → E defined on a subset J of an interval [a, b] in R with a, b ∈ J that is an isometry with g(a) = x and g(b) = y which cannot be isometrically extended to some larger subset of [a, b] as a map with values in E. It follows from Zorn's Lemma that such geodesics always exist. For more details and generalizations, see Definition 4.1.
In general, in accordance with the usage of this concept in metric geometry, a geodesic need to have the shortest distance property only locally, see Definition 6.1.
The reason why both of these aspects-the investigation of the group T E on the one side and (shortest) geodesics in arbitrary metric spaces on the other side-are now brought together is as follows: For many metric spaces M = (E, d), for which there exists only one geodesic trace between any two distinct points-which will be called unigeodesic spaces, the structure of the group T E can be determined completely. This holds in particular if, in addition, the given metric space is a tripod space as studied in Joharinad and Jost (2019). Tripod spaces are also studied in Bandelt (1993) in terms of modular interval spaces. We should point out, however, that in general tripod spaces, shortest geodesics need not be unique, and thus, the unigeodesic property need not hold. For those unigeodesic spaces, which are in addition tripod spaces, it does not matter whether the given metric space is discrete or topologically connected. This is particularly interesting by studying trees-discrete ones or their geometric realizations.
In particular, any hyperconvex vector space is a tripod space. Hyperconvexity means the following Helly property: Whenever (B i ) i∈I is a family of closed balls such that any two of these intersect, then all of these balls have a nonempty intersection.
Examples are spaces of bounded functions with the supremum norm. This is interesting within our context, because Kuratowski proved in Kuratowski (1935) (a similar idea can already be found in Fréchet 1906) that every metric space M = (E, d)-of arbitrary cardinality-may be isometrically embedded into an L ∞ -space, which is hyperconvex (see Isbell 1964). Again, however, this hyperconvex hull is in general not unigeodesic, as it carries an L ∞ -type geometry. For finite E, see also Dress (1984).
Technically, the paper is organized as follows: In Sect. 2, we study arbitrary intervals and betweenness relations. In Sect. 3, we introduce the group T E for any nonempty set E with a betweenness relation. Furthermore, we prove first results. Some of these are analogous to corresponding results about the Tutte group of a matroid.
In Sect. 4, we introduce shortest geodesics in arbitrary metric spaces and present examples. The general concept of a geodesic will be discussed in Sect. 6.
In Sect. 5, further structural results about the group T E -for metric spaces M = (E, d)-are proved.
Finally, we want to remark the following: Although (complete) ordered sets and metric spaces are-in general-quite different structures, elementary results about the group T E for ordered sets will be used repeatedly to obtain deeper results for metric spaces-in terms of their geodesics, because any geodesic may be equipped with two inverse complete orderings, which yield the same betweenness relation as that given by the metric.

Intervals and betweenness relations
For a given nonempty set E, we consider abstract intervals [a, b] ⊆ E for a, b ∈ E satisfying the following conditions: Example 2.1 For an ordered set (X , ≤) and a, b ∈ X put
Note that these two examples studied so far are-in general -quite different. However, if the set R of real numbers is equipped with the usual order or the usual metric, all intervals in these examples coincide! This holds also if the set R is replaced by any nonempty subset E.

Remark 2.3 Assume that conditions
Remark 2.6 Assume that a betweenness relation is given on a nonempty set E and that a, b, c ∈ E are pairwise distinct. Then we have:

Convention 3.4 For a nonempty set E and an abelian group G, the abelian group G E f in
consists of all mappings f : For a singleton set {a}, we write also δ a instead of δ {a} .

Proposition 3.5
Suppose that E is a nonempty totally ordered set and that e 0 ∈ E is fixed. Put E := E \ {e 0 }. Then the Dress group T E is freely generated by all elements T e 0 ,e for e ∈ E . That means: Proof Clearly, all elements T e 0 ,e for e ∈ E generate T E , because for different elements e, f ∈ E one has Hence, ϕ induces a homomorphism ψ : Assume that e 1 , ..., e n ∈ E are pairwise distinct and that k 1 , ..., k n ∈ Z are such that n i=1 k i · T e 0 ,e i = 0. We must still verify: k i = 0 for 1 ≤ i ≤ n. Now we make use of the map ψ: There exist s 1 , ..., s n ∈ {−1, 1} such that Since all n + 1 elements e 0 , e 1 , ..., e n are pairwise distinct, we obtain at once k i = 0 for 1 ≤ i ≤ n as claimed.
Next we study metric spaces M = (E, d)-with the betweenness relation, induced by the metric d-and group homomorphisms from T E into better known abelian groups as well as group homomorphisms from T E 1 to T E 2 between two appropriate metric spaces In particular, the group T E cannot be finite-if E has at least 2 elements.
Proof Consider the well defined homomorphism g : F E → R defined by g(X a,b ) := d(a, b). It follows from the very definition of K E that K E is contained in the kernel of g. This proves what we want.
We get now also the following almost trivial but significant Corollary 3.7 Suppose that a, b, c are three pairwise distinct points in a metric space M = (E, d). Then the following two statements are equivalent: If f is as in Proposition 3.6, then (I I ) implies: Furthermore, we have the following Proposition 3.8 Suppose that E 1 and E 2 are sets with betweenness relations * , , respectively, and that ϕ : where the generators of T E 1 are denoted by T a,b -instead of T a,b . Note that this statement holds particularly if (E 1 , d 1 ) and (E 2 , d 2 ) are metric spaces and if ϕ : E 1 → E 2 is an isometric embedding.
Proof Consider the well defined homomorphism h : This means that K E 1 is contained in the kernel of h as claimed.
The following example shows in particular that the homomorphism ψ as just studied is not necessarily injective-what is in particular the reason for the notations T a,b within T E 1 .
Example 3.9 Suppose that M 1 = (E 1 , d 1 ) is a -finite-discrete metric space of cardinality n := |E 1 | ≥ 2, with the metric d 1 (a, b) = 1 for all a, b ∈ E 1 with a = b. Hence, there do not exist any three pairwise distinct elements a, b, c ∈ E 1 satisfying b ∈ [a, c]. That means: The group T E 1 is freely generated by all elements T a,b for any two-element set {a, b} ⊆ E 1 , whence T E 1 is isomorphic to Z 1 2 ·n·(n−1) . Now consider a new point z outside of E 1 as well as the set E 2 := E 1 ∪ {z} and the wheel G with vertex set E 2 , whoose edges are precisely the two-element sets {z, e} for e ∈ E 1 . By considering the weight function w defined on the edges of G given by w({z, e}) = 1 2 for all e ∈ E 1 , the graph G becomes a metric space M 2 defined on E 2 with M 1 as an isometric subspace. Now, if n ≥ 4, choose 4 pairwise distinct elements a, b, c, d ∈ E 1 . Then-as pointed out before-we have Hence, ψ is not injective.-More precisely, we see that T E 2 is isomorphic to Z n , because the wheel G has precisely n edges.
Consider in addition the case n = 3, say E 1 = {a, b, c}. Then both of the groups T E 1 and T E 2 are-as abstract groups-isomorphic to Z 3 , but ψ is not an isomorphism: The three canonical generators T a,z , T b,z , T c,z ∈ T E 2 do not lie in the image of ψ (what follows easily, in particular together with Proposition 3.6).-But all three sums of two distinct generators lie in the image of ψ, whence the factor group T E 2 /ψ(T E 1 ) has order 2. Altogether, it follows-for n = 3, that ψ is not surjective but injective. This example will become interesting again at the end of this paper -by studying more general trees as well as isometric subspaces of these trees.
At the end of this section, we want to point out the following trivial but very useful proposition-which follows by applying Proposition 3.8 twice: Proposition 3.10 Suppose that E and E are sets with betweenness relations , * , respectively, and that f : As we shall see later, this proposition is not only valuable if f is a-surjectiveisometric map between metric spaces.

Shortest geodesics in arbitrary metric spaces
To study the Dress group T E for general metric spaces M = (E, d) more exhaustively, we give the following general definition of shortest geodesics. (i) For x, y ∈ E, a shortest geodesic from x to y is a map g : J → E defined on a subset J of a compact interval [a, b] ⊆ R with a, b ∈ J and satisfying the following conditions: that means, g is an isometric map. (G2) There does not exist an isometric continuationg : (g n ) n∈N of geodesics g n : J n → E from points x n to y n , respectively, such that every geodesic g n is a restriction of g n+1 and of g and such that J is the union all of these sets J n .
If this map g is a shortest geodesic from a point x to a point y, we say also that G is a geodesic trace between x and y. (v) A geodesic line l in E is a maximal geodesic trace. That means: l is a geodesic trace, but there does not exist any geodesic tracel that properly contains l. and g 2 : J 2 → E are shortest geodesics from x to y or from y to z, respectively, with t 0 := max(J 1 ) = min(J 2 ) and, hence, g 1 (t 0 ) = y = g 2 (t 0 ), then the common continuation g : J 1 ∪ J 2 → E of g 1 and g 2 is clearly a shortest geodesic from x to z: All t 1 ∈ J 1 and all t 2 ∈ J 2 satisfy t 2 − t 1 = d(g 1 (t 1 ), y) + d(y, g 2 (t 2 )) = d(g 1 (t 1 ), g 2 (t 2 )),

Remark 4.2 Assume that
and there does not exist any proper isometric continuation of g from x to z and with values in E, because g 1 and g 2 are maximal-with values in E, too.
(iv) If x, y ∈ E are arbitrary, then Zorn's Lemma implies that there exists at least one shortest geodesic from x to y: Namely, assume that (h i : J i → E) i∈I is a chain of certain isometric maps with x, y ∈ h i (J i ) ⊆ [x, y] for all i ∈ I and such that any map h i is a restriction of h j whenever J i ⊆ J j . Then we get a well defined isometric map h from the union J all of these sets J i into [x, y] such that h is a continuation all of the maps h i . (v) Assume again that x, y ∈ E are arbitrary-and that, in addition, x 0 , x 1 , ..., x n ∈ E are pairwise distinct elements with x 0 = x, x n = y and n i=1 d(x i−1 , x i ) = d(x, y). Then it follows also that there exists at least one shortest geodesic from x to y whose trace contains all points x 0 , ..., x n : Either modify the above consideration by looking only at chains of isometric maps having all elements x 0 , ..., x n in their images.-Or apply iv) to construct shortest geodesics from x i−1 to x i for all i with 1 ≤ i ≤ n. Then these n shortest geodesics together give rise to one shortest geodesic from x to y; just apply iii)repeatedly. (vi) Assume now that (h i : J i → E) i∈I is a chain of arbitrary shortest geodesics in E-and arbitrary (infinite) cardinality of I . We conclude that this chain has a supremum which is again a shortest geodesic in E: We get once more a well defined map h : J → E defined on the union J all of the sets J i such that any h i is a restriction of h.
For n ∈ N, put a n := a if a ∈ J . Otherwise, let (a n ) n∈N denote a descending sequence in J which converges to a. Similarly, put b n := b if b ∈ J . Otherwise, let (b n ) n∈N denote an ascending sequence in J which converges to b. In addition, we assume that a 1 ≤ b 1 . Finally, we can now choose g n as the restriction of h to J ∩ [a n , b n ] for n ∈ N-and see that h is the supremum of the chain (g n ) n∈N . Hence, by Definition 4.1 ii), we conclude that h is a geodesic in E. (vii) Zorn's Lemma implies now directly-together with vi) -that any shortest geodesic in E is contained in a maximal shortest geodesic. Accordingly, any geodesic trace in E is contained in a geodesic line in E.
The number n is the length of this path. Then we get a metric defined on V by denoting by d(v, w) the length of a shortest path from v to w. The shortest geodesics from v to w are precisely the shortest paths from v to w.

Remark 4.4 By Isbell (1964), every metric space M = (E, d) may be isometrically embedded into a Banach space (V ,d)of bounded functions with the supremum norm. This embedding is isometric in the strong sense that the restriction ofd to E coincides with d.
We point out that any shortest geodesic g : J → E for a subset J of R with at least two elements may be continued to a shortest geodesicg : R → V -within several steps: (i) First of all, g can be-uniquely-continued to a shortest geodesic g 1 : cl(J ) → V , defined on the topological closure J 1 := cl(J ) of J , because V is a Banach space. (ii) Let J 2 denote the smallest-bounded or unbounded-interval containing J 1 ; hence, we have J 2 = J 1 if and only if J 1 is connected. Put g 2 (t) := g 1 (t) for t ∈ J 1 . If q ∈ J 2 \ J 1 , then there exist uniquely determined elements x 0 , y 0 ∈ J 1 with x 0 < q < y 0 such that the open interval (x 0 , y 0 ) does not intersect the closed set J 1 . Then put This means: g 2 is affine linear on the interval [x 0 , y 0 ]. This way, we get a geodesic g 2 : J 2 → V that continues g 1 . (iii) If J 2 = R, nothing more needs to be done.
If J 2 = [a, b] is a compact interval with a < b, there might be several possibilities to continue g 2 onto all of R. As we cannot suppose that g 2 is affine linear on J 2 , we proceed as follows-by making repeatedly use of point mirrorings: Put d := b − a, and defineg : R → V by: if t ∈ J 2 , and k ∈ Z is even, if t ∈ J 2 , and k ∈ Z is odd.
Note that for arbitrary l ∈ Z, both of these instructions yield-in view of l · d + a = (l − 1) · d + b : This way we get a geodesic line in the vector space V , because it is point symmetric with respect to any pointg(l · d + a) for l ∈ Z. Instead, we may-still in case J 2 = [a, b]-proceed as follows, what is indeed simpler, but one might ask which version is aesthetically better: Put If, on the other side, J 2 has a maximum or a minimum, but not both, say J 2 = [a, ∞), then the considerations become simpler: One just putsg Theng is a shortest geodesic becauseg(a) = g 2 (a) is betweeng(a − t) and g(a + t) for any t > 0. Thus, in general, there is only one point mirroring that sends the trace ofg onto itself.

Further structural results
In this section, we are mainly interested in further results about metric spaces whose geodesics show a specific nature.

Definition 5.1 Suppose that M = (E, d) is a metric space with at least two elements.
(i) The metric space M = (E, d) is called unigeodesic, if for any two distinct points x, y ∈ E there exists a unique geodesic trace from x to y.
In this case, we put also x y := [x, y]-which is just the unique geodesic trace from x to y. (ii) The metric space M = (E, d) is unilinear, if for any two distinct points x, y ∈ E there exists a unique geodesic line l that contains x and y.
Remark 5.2 Informally, a space is unigeodesic when there is a unique way to interpolate between any two distinct points, and it is unilinear if extrapolation is unique.

Remark 5.3
Any unilinear metric space is unigeodesic, because any two distinct geodesic traces between two different points x, y are contained in different geodesic lines. If, furthermore, M = (E, d) is a unigeodesic or unilinear metric space, then any subspace M = (E , d ) is unigeodesic or unilinear too, respectively, because for any two points A, B ∈ E , any two distinct geodesic traces G 1 , G 2 ⊆ M between A and B are contained in distinct geodesic traces as well as in distinct geodesic lines in M.
Furthermore, we observe the following Examples 5.4 Euclidean and hyperbolic spaces are unilinear. If M = (E, d) is a-topologically connected or discrete-tree, then M is unigeodesic but not unilinear, if M has at least one branching point.
We can now prove the following

Proposition 5.5 Suppose that the metric space M = (E, d) is unilinear. Then the Dress group T E is free abelian. More precisely, we have: (i) For any geodesic line l in M, consider the metric subspace M l of M defined
on l, and fix some point x l ∈ l. Then the group T l is freely generated by all elements T x l ,x for x ∈ l \ {x l }. (ii) T E equals the direct sum over all groups T l for all geodesic lines l in M.
It follows that T E is freely generated by all elements T x l ,x for all geodesic lines l and all elements x ∈ l \ {x l }.
Proof (i) follows directly from Proposition 3.5, because every geodesic line l can be totally ordered-in two obvious ways-such that the betweenness relation defined by the order equals the betweenness relation induced on the geodesic trace l. (ii) The second assertation follows at once from the first one and i). Concerning to the first assertion, we get a well defined and canonical epimorphism ϕ from the direct sum over all groups F l into T E just by mapping any X Next we want to study the group T E for unigeodesic metric spaces M = (E, d) more thoroughly.
First we prove the following more general.

Proposition 5.6 Suppose that M = (E, d) is a metric space and that T is a nontrivial torsion element in T E , say n · T = 0 for some n ∈ N with n ≥ 2. Then there exists a finite subspace M = (E , d ) of M such that T = ψ(T ) for some -nontrivialtorsion element T ∈ T E , where ψ is the canonical homomorphism from T E to T E as in Proposition 3.8. In other words: If T E is torsion-free for any finite subspace M = (E , d ) of M = (E, d), then T E is torsion-free, too.
Proof If ν : F E → T E denotes the canonical epimorphism-as in Convention 3.2, then there exists an element Z ∈ F E with ν(Z ) = T -and, hence, n · Z ∈ K E . This means: n · Z is a sum of finitely many elements of the shape If ν denotes the canonical epimorphism from F E onto T E and T := ν (Z ), then T is a torsion element in T E , and we get T = ν(Z ) = ψ(T ) for ψ : T E → T E as in the proposition.
Concerning the group T E for finite metric spaces, we formulate the following

Remark 5.7 Suppose that the metric space M = (E, d) is finite and that
Note that every T ∈ T E \ {0} admits a reduced presentation by the very definition of T E , because E is finite. Now we can prove the following Theorem 5.8 For any unigeodesic metric space M = (E, d), the Dress group T E is torsion free.
Proof By Proposition 5.6 and Remark 5.3, we can assume that E is finite. Assume that is a nontrivial torsion element in T E and that the given presentation is reduced. Let u denote a-small-positive real number that is smaller than all differences of the form Since E is finite, such a number u exists. Now define a new metric d on E by Then d is again a metric with the same betweenness relation that is induced by d; in the following verification, intervals will mean intervals in M.
On the other hand, assume now d(  d (A, B).
Since T is a torsion element, and R is torsion free, we get but also This is of course not possible, because k 1 = 0 and u > 0.
Next we look at the particular case of graphs.
Convention 5.9 Suppose that n ≥ 3. In what follows, the cyclic graph C n with n vertices has vertex set E = V n := {v 1 , ..., v n } and edge set where we write v 0 := v n for simplicity.
Example 5.10 Assume that n ≥ 3 is odd, say n = 2k + 1 for some k ∈ N. Then the metric space M = (E, d) defined by C n is given by This metric space M is unigeodesic. If (A 0 , ..., A j ) is a path in C n with j ≤ k (and pairwise distinct vertices), then this path is also the uniquely determined shortest geodesic between A 0 and A j . Hence we have the reduced presentation This implies that T V n is isomorphic to Z n .

Example 5.11
If n = 2k is even for some k ≥ 2, the situation changes: As before, if (A 0 , ..., A j ) is a path in C n with pairwise distinct vertices, now with j < k, then this path is again the uniquely determined shortest geodesic between A 0 and A j .-But if A and B are opposite vertices in C n , then there exist two shortest geodesics between A and B.
As v 0 is opposite to v k , we get The last two equations imply-by subtraction: This means: The element is either the neutral element in T E or has order 2. By symmetry, the same holds for all differences But all these elements have indeed order 2: Let e 1 , ..., e k denote the characteristic generators of Z k ; these are also the canonical unit vectors in Q k . If, moreover, 1 , ..., k−1 build a basis of the vector space (Z/(2 · Z)) k−1 , then we get an isomorphism ϕ : T V n → Z k × (Z/(2 · Z)) k−1 given by Note that we cannot involve k independent elements of order 2 to ensure that the first equation stated above-and analogous equations concerning any two different geodesics of length k -hold.
Summarizing, we see that T V n is isomorphic to the group Z k × (Z/(2 · Z)) k−1 .
Example 5. 12 We study again the cyclic graph C 4 , but now with a modified metric: The distances of adjacent vertices are defined as follows: Now, there are again two shortest geodesics between v 0 and v 2 -but now of length 4, namely (v 0 , v 1 , v 2 ) and (v 0 , v 3 , v 2 ). But there is only one shortest geodesic between v 1 and v 3 -of length 3, namely (v 1 , v 0 , v 3 ).
admit essentially only one nontrivial relation, namely This implies that T V is isomorphic to Z 3 . This example shows also that the converse of Theorem 5.8 does not hold.
Example 5. 13 We study one more graph G, now with vertex set V = {v 0 , v 1 , v 2 , v 3 , v 4 } and edges {v 0 , v i } for 1 ≤ i ≤ 3 as well as {v i , v 4 } for 1 ≤ i ≤ 3. With the induced metric, the cyclic graph C 4 can be imbedded into G in several ways. By Example 5.11 and a repeated application of Proposition 3.8, we conclude that all of the differences Moreover, by Proposition 3.6, the group T V is not finite and, hence, isomorphic to some group of the form Z × (Z/(2 · Z)) r for some integer r .-This holds for r = 3: If 1 , 2 , 3 are independent elements of order 2, then we get a well defined isomorphism ϕ : T V → Z × (Z/(2 · Z)) 3 given by Note also: This example yields-beyond Example 3.9-a new constellation where the homomorphism constructed in Proposition 3.8 is not injective.
Concerning Example 5.11 in case n = 4, we want to formulate an even more abstract version-which will be useful afterwards by studying R 2 , endowed with the Manhattan metric.
Proposition 5.14 Suppose that E = {a, b, c, d} is a set with four elements, together with that betweenness relation which is uniquely determined by Then the Dress group T E is isomorphic to Z 2 × (Z/(2 · Z)). In particular, the elements coincide. Moreover, T 1 = T 2 has order 2..

Proof
The cyclic graph C 4 as studied in Example 5.11 exhibits exactly the same betweenness relation as the set E given here; just It was verified in Example 5.11 that T 1 and T 2 coincide-and that this element has order 2.
Hence, the result follows now-by applying also Proposition 3.10.
As a further application, we now consider the metric space M 1 defined on R 2 , together with the Manhattan Metric d 1 ; that means d 1 ((x, y), (u, v)) = |x −u|+|y−v| for all x, y, u, v ∈ R.
As a convention, we write now T M 1 instead of T E .
We first prove the following Lemma 5.15 Suppose that x, y ∈ R * , and put Then one has T 1 = T 2 , and this element has (at most) order 2.
Proof This result follows now at once from Proposition 3.10 and Proposition 5.14, because the four points a = (0, 0), b = (x, 0), c = (x, y), d = (0, y) satisfy the requirements stated in the last named proposition.
Now we can prove the following result, where, on the one side, the last lemma will be used, but, on the other hand, we shall see also that the elements T 1 = T 2 examined there have indeed order 2.
Then ϕ induces an isomorphism ψ : Proof First of all, note that f is well defined-as a homomorphism -by Lemma 5.15.
Next, observe that the group K M 1 is generated by all elements as follows: Concerning the generators listed in I I a) and I I b), we may assume that 0 ∈ {x 1 , x 2 , x 3 } or 0 ∈ {y 1 , y 2 , y 3 }, respectively. Then it follows from the definition of ϕ that ϕ(X ) = 0 holds for all elements X as listed in I ), I I a), I I b). Moreover, it follows from the definition of ϕ-and the verification at the beginning of this proof-that the same holds for all elements X as listed in I I I a) and I I I b).
Hence ϕ induces at least a well defined epimorphism ψ : T M 1 → G 1 × G 2 . It remains to prove: f (ψ(T )) = T for all T ∈ T M 1 .
Note that T M 1 is generated by the set It follows at once from the definitions that f (ψ(T )) = T holds at least for all T ∈ A. This proves what we want.
Similar results can be proved for the Manhattan Metric defined on R n for n > 2; however, the technical details are again much more complicated. Now we return again to unigeodesic metric spaces-with one more specified property as studied in Joharinad and Jost (2019) Remark 5.18 Definition 5.17 means that there are three geodesic traces between x, y, between x, z and between y, z, which have at least one point in common.-It is not required that the point m is uniquely determined, but this follows automatically if M = (E, d) is a unigeodesic space.
It follows at once that any point m as in the definition satisfies We now prove the following result which is analogous to Proposition 3.5.
Then ϕ induces an isomorphism ψ from the Dress group T E onto the subgroup H of Z E f in that is generated by all differences δ A − δ A 0 for A ∈ E .
Proof Clearly, we have ϕ(F E ) = H . It remains to prove that K M is the kernel of ϕ. Suppose that A, B, C ∈ E satisfy B ∈ [A, C]. If all four (not necessarily pairwise distinct) points A 0 , A, B, C lie on a common geodesic trace G, then -as in Proposition 3.5-it follows that X A,C − X A,B − X B,C lies in the kernel of ϕ, because G may be viewed as a totally ordered set with the same betweenness relation as that induced by the metric d. If as claimed.
Now, it follows that ϕ induces an epimorphism ψ : T E → H . It remains to verify that ψ is injective.
Consider (again) the relations as well as These relations imply that T E is generated by all elements of the shape T A 0 ,A for A ∈ E . By the definitions of ϕ and ψ, it follows at once that ψ is injective.
One important special case of the last result is as follows: Example 5.20 Serre and Bass (1977) identified trees as representation spaces for the linear algebraic group SL 2 over p-adic fields. In this context, the notion of an R-tree was introduced by Tits (1977). For systematic treatises of the topic, see for instance Morgan (1992), Bestvina (2002). R-trees also play an important role in the work by Andreas Dress. A metric space (T , d) is an R − tree, if the following two axioms hold: (T1) For all p, q ∈ T there exists a uniquely determined isometry ψ : [0, d( p, q)] → T such that: ψ(0) = p, ψ(d( p, q)) = q. Remark 6.2 • We may also define more generally geodesic nets (see Jost and Todjihounde 2007) along these lines. • We can formulate the definition of a (closed) geodesic simply in terms of a betweenness relation, without the need for a metric. We only need to require g(t i−1 ) g(t i ) g(t i+1 ) for i = 1, . . . , n − 1 in place of (GG1b) in the preceding.

Example 6.3
The cyclic graphs C n for n ≥ 4 defined above are closed geodesics.
Example 6. 4 We add to a cyclic graph C n with n = 2k, k ≥ 4 vertices w 1 , . . . , w k and connect each w j with v j and v j+k . Two vertices connected by an edge in the resulting graph G n have distance 1, and the distance function is extended in the usual way to other pairs of vertices. Then the vertices of the cycle C n still form a closed geodesic within G n , but this closed geodesic no longer generates elements of order 2 in the Dress group, because no pair of vertices in C n then is connected by two shortest geodesics that are entirely contained in that cycle, as we can always find some shorter curves going through some w j . G n is not a unigeodesic metric space, as, for instance, there exist two shortest geodesics from w 1 to w 2 , namely (w 1 , v 1 , v 2 , w 2 ) and (w 1 , v k+1 , v k+2 , w 2 ).

Concluding remarks
In future research, it might be interesting to study the Dress group for further metric spaces that are important in several fields of geometry, like spheres and projective spaces. Moreover, given any unitary ring R, one might ask whether an examination of the free R-module T E (R) may be of interest as well-this way abstracting from the specific ring Z to a general ring R.