Fields Institute Communications Volume 00, 0000

The arithmetic–arboreal residue structure of a Prüfer domain, I

Şerban A. Basarab

Institute of Mathematics “Simion Stoilow” Romanian Academy P.O.Box 1–764, RO-70700 Bucharest, ROMANIA

                                Serban.Basarab@imar.ro

Abstract. The paper is the first of a series devoted to a systematic study of the residue structure of a Prüfer domain with a stress upon the intimate connection between its arithmetic and arboreal properties.

Introduction

It is well known that, with few exceptions, quantifier elimination for the theory of value groups and for the theory of residue fields is not enough to guarantee quantifier elimination for given theories of henselian valued fields. Thus we are led in a natural way to the question whether there exist suitable more reach residue structures doing such a job. Such residue structures called mixed structures were introduced in [3] as an adequate tool for proving a theorem on relative elimination of quantifiers for henselian valued fields of characteristic 0, which extends the wellknown results of Macintyre [18] and Prestel-Roquette [20] on p-adically closed fields. A consequence of a purely algebraic nature, namely an isomorphism criterion for henselian valued fields of characteristic 0, algebraic over a common valued subfield, is presented in [11]. The results of [3] and [11] are reconsidered in a larger framework in [17] by using the notion of tame extension. Note also that the mixed structures above are suitably modified in [22] to get quantifier elimination for a class of D-henselian fields.

Roots of such mixed residue structures reflecting the additive as well as the multiplicative structure of a valued field can be traced in classical papers of Hasse and Krasner. Notice that structures of this type, namely Krasner’s multiplicative quotient hyperfields of valued fields, occur in a special very important case in Deligne’s paper [14].

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

1991 Mathematics Subject Classification. Primary 13F05, 05C05; Secondary 12J10, 06F20.

The author was supported in part by grants offered by the organizers of the Conference and by the Romanian Academy.

On the other hand, having a different motivation, in the work [8] devoted to generalized trees as compatible families of “semiracks”, I sketched the possibility of extending to Prüfer domains Tits’ construction [23] of the tree associated to a valuation ring. Meanwhile, in the works [4–7, 9, 10], I made a systematic study of the generalised trees with some applications to the combinatorial group theory.

As it is well known, in the last years various types of trees have been the subject of much investigation mixing very naive geometric considerations with sophisticated geometric and algebraic structures and techniques. A natural generalization of simplicial trees (i.e. acyclic connected graphs) was introduced in the fundamental paper [19] by Morgan and Shalen under the name of Λ–trees. This notion is obtained from that of simplicial tree, interpreted in a natural way as metric space with an integer–valued distance function, by replacing ℤ by any totally ordered abelian group Λ. Tits’ construction connecting simplicial trees and SL_{2} of a local field is extended in [19] to arbitrary valued fields. Namely, if K is a valued field with valuation v : K^{∗} → Λ := vK and if G(K) is a rank-1 algebraic group, as for instance SL_{2}(K), then there is a Λ–tree on which G(K) acts.

In their study of group actions on Λ–trees, Alperin and Bass [2] regarded the fundamental problem of this subject to be “to find the group- theoretic information carried by a Λ–tree action, analogous to that presented in (Serre’s book) Trees for the case Λ = ℤ.” In the papers [4–6] concerned with the problem above, I developed a technique having two complementary aspects : a group theoretic one concerning group actions on groupoids and a metric one concerning length functions on groupoids taking values in a lattice ordered abelian group Λ, with the aim to find suitable generalizations for the notions of ordinary trees and graphs, graph of groups, universal covering relative to a graph of groups, etc. Thus, in [6], based on the the wellknown fact that in a simplicial tree (or more generally in a Coxeter complex) for any three vertices x, y, z there exists a unique vertex Y (x, y, z) lying on the geodesics connecting any two of them, I defined a very general notion of tree (called Y–tree in [6]) as a set X together a ternary operation Y : X^{3} → X subject to three equational axioms (see 1). These generalized trees have nice algebraic and geometric properties, to mention only a Stone-like duality theorem ([7], [21]) and the fact that two basic operators on the class of generalized trees assigning to a generalized tree X the generalized tree Dir(X) of the directions on X, respectively the generalized tree F old(X) of the foldings(or retractions) of X, commute ([9]).

To my surprise I learned quite recently that this general notion of tree has been known for a long time under the name of median algebra, mainly in the context of universal algebra (see [12, 16, 24]). Moreover it seems that median algebras are very little known amongst group theorists and only recently attracted their interest. Let’s mention here Roller’s work [21] where the median algebras are used for an extended study of Dunwoody’s construction and Sageev’s theorem on cubings.

The mixed structures from [3] can be suitably generalized to Prüfer domains and in their investigation the notions and methods of the arboreal group theory play an important role. The present paper is the first one from a series devoted to a systematic study of the residue structures associated to a Prüfer domain with a stress upon the close relation between their arithmetic and arboreal properties. This study is motivated by further applications to the model theory of rings satisfying a local-global principle and to the arboreal structure of SL_{2}(K), where K is the field of fractions of a Prüfer domain, in particular of a Dedekind domain.

To give the reader some idea on our approach, let us consider a valued field K with valuation v, valuation ring A, residue field K_{v} and value group vK. Denote by Γ the product vK × vK with the lexicographic order (α, α′) ≤ (β, β′) iff either α < β or α = β and α′ ≤ β′. Set Γ_{+} := {(α, α′) ∈ Γ|(α, α′) ≥ (0, 0)}.

Given (α, α′) ∈ Γ and x, y ∈ K, write x ≡ y mod (α, α′) if v(x − y) > α for α′ > 0, resp. v(x − y) ≥ α for α′ ≤ 0, and let K/(α, α′) denote the corresponding quotient A–module. Denote by X := X(A) the disjoint union of the A–modules K/(α, α′) for (α, α′) ∈ Γ. First notice that the field operations of K induce an arithmetic structure on X trough the naturally defined operations

x mod(α, α′) + y mod(β, β′) = (x + y) mod min((α, α′), (β, β′))

and

x mod(α, α′) • y mod(β, β′) = xy mod(γ, γ′),

where

(γ, γ′) = min((α + β, α′ + β′), (α + v(y), α′), (β + v(x), β′))

W.r.t. the operations above, X becomes an additive, resp. a multiplicative, commutative inverse semigroup, and the mapping Γ → X, (α, α′) ↦ 0 mod(α, α′), resp. Γ_{+} → X, (α, α′) ↦ 1 mod(α, α′), maps injectively Γ, resp. Γ_{+}, onto the idempotents of (X, +), resp. of (X, •). Note that ϵ := 0 mod(0, 0) = 1 mod(0, 0) is the only element of X which is an idempotent w.r.t. + as well as to •. For x = x mod(α, α′) ∈ X, write e_{+}(x) = (α, α′) ∈ Γ, and v(x) := e_{+}(ϵ • x) = min((v(x), 0), (α, α′)). Notice that the mapping v : X → Γ is a surjective morphism from the commutative semigroup (X, •) onto the abelian group (Γ, +) satisfying the valuation-inequality v(x + y) ≥ min(v(x), v(y)). The equivalence relation induced by v is the minimal congruence on (X, •) for which the quotient semigroup is a group.

On the other hand, X has a canonical structure of Γ–tree w.r.t. the distance function d : X × X → Γ_{+}, (x, y) ↦ e_{+}(x) + e_{+}(y) − 2v(x − y). The Γ–tree X is directed w.r.t. the natural partial order on X given by x ≤ y iff x = y + e_{+}(x), i.e. e_{+}(x) ≤ e_{+}(y) and x ≡ y mod e_{+}(x).

The arithmetic and the arboreal structures on X as defined above are closely related. A detailed study of these structures, including an axiomatic approach of them, is the object of a forthcoming paper devoted to the more general case of a quasi-boolean Prüfer domain A.

To convince the reader on the usefulness of this approach, let us rephrase in the terms above the main result(Theorem A) of [3] on relative elimination of quantifiers for henselian fields of characteristic 0. With the notations above, let’s assume that char K = 0 and char K_{v} = p ≠ 0. For k ∈ ℕ, let X_{k} := {x mod(α, α′) ∈ X | α − v(x) ≤ kv(p)}. The X_{k}’s form an ascending chain of substructures of the full residue structure X. In suitable first order languages, the X_{k}’s are interpretable in the mixed k-structures of [3] and viceversa, so we can state

Theorem A’. Given the valued field extensions L/K and F/K, where char K = 0, char K_{v} = p > 0, and L, F are henselian, the necessary and sufficient condition for the valued fields L, F to be elementarily equivalent over K is that for all k ∈ ℕ, the corresponding arithmetic-arboreal residue structures X(L)_{k}, X(F)_{k} are elementarily equivalent over X(K)_{k}.

The present paper is organized in two sections. The first one having a preliminary character introduces the reader to the basic notions and facts to be used later. There are four subsections devoted to median sets (or generalized trees), convexity and the prime spectrum of a median set, directions and foldings on median sets, Λ-metric median sets (where Λ is a lattice ordered abelian group), and Λ-valued lattices. Since the proofs are in general omitted or only sketched, the interested reader may consult for details the works [6, 7, 9, 21, 24].

The second section entitled “Arboreal structures on Prüfer domains” having seven subsections is devoted to some basic properties of the lattice L(A) of the finitely generated A-submodules of V := K^{2} spanning the two-dimensional vector space V, where K is the field of fractions of a Prüfer domain A, the symmetric Λ–forms on the lattice L(A), where Λ ≅ D(A) is the lattice ordered abelian group of divisors of K, the “index” (M : N) for M, N ∈ L(A), the Λ–metric median set structure on the space X(A) := D(A)\L(A) of D(A)–orbits of L(A), the description of X(A) as a collection of affine subsets of K(cosets of elements of K modulo divisors of K), the action of GL(V) on the Λ-metric median set X(A), and some remarkable foldings on X(A) and their stabilizers in GL(V).

1 Preliminaries on median sets (generalized trees)

By a median set (or a generalized tree) we understand a set X together with a ternary operation Y : X^{3} → X satisfying the following equational axioms:

(M1) Symmetry : Y (x, y, z) = Y (y, x, z) = Y (x, z, y)

(M2) Absorptive law : Y (x, y, x) = x

(M3) Selfdistributive law : Y (Y (x, y, z), u, v) = Y (Y (x, u, v), y, Y (z, u, v)) The median sets form a category with naturally defined morphisms, arbitrary

direct products, an initial object - the empty median set - and a final object - the singleton.

The distributive lattices (not necessarily bounded), in particular, the l.o. abelian groups, have a canonical underlying structure of median set w.r.t. the operation

Y (x, y, z) := (x ∪ y) ∩ (y ∪ z) ∩ (z ∪ x) = (x ∩ y) ∪ (y ∩ z) ∪ (z ∩ x)

Notice that for any a ∈ X, the binary operation (x, y) ↦ ∨_{a} := Y (x, a, y) makes X a sup-semilattice with the last element a; denote by ≤_{a} the corresponding partial

order.

It is well known that the median sets are closely related to the so called betweenness relations. More precisely, given a set X, the mapping assigning to a ternary operation Y on X the ternary relation R_{Y} : xyz ↔ Y (x, y, z) = y maps bijectively the median sets with support X onto the ternary relations on X satisfying

(i) R_{Y} is a betweenness relation, i.e. xxy, xyx → x = y and xyz ∧ xuy → zux for all x, y, z, u ∈ X, and

(ii) for all x, y, z ∈ X there exists a unique element u(= Y (x, y, z)) such that xuy, yuz and zux.

1.1 Convexity and the prime spectrum of a median set. For x, y ∈ X, set [x, y] = {z ∈ X | Y (x, y, z) = z} = {Y (x, y, z) | z ∈ X}.

A subset I of a median set X is said to be convex if [x, y] ⊆ I whenever x and y belong to I.

In particular, any convex subset I of X is closed under the operation Y, so it is a median subset of X. As the intersection of a family of convex subsets of X is convex too, we may speak on the convex closure of (or the convex subset generated by) any subset A of X and denote it by [A]_{X}, omitting usually the subscript X. Note that [∅] = ∅, [{a}] = {a} for a ∈ X, and [{a, b}] = [a, b] for a, b ∈ X.

By a cell (or a simplex) of the median set X we understand a convex subset of the form [a, b] with a, b ∈ X. Given a cell C of X, any a ∈ X for which there exists b ∈ X such that C = [a, b] is called an end of the cell C. The (non-empty) subset of all ends of a cell C, denoted by ∂C, is called the boundary of C.

The boundary ∂C of a cell is a median subset of C and there exists a canonical automorfism ¬ of ∂C such that C = [a, ¬a] and ¬¬a = a for all a ∈ ∂C. Given a ∈ ∂C, the cell C becomes a bounded distributive lattice under the order ≤_{a},

with the join ∨_{a}, the meet ∨_{¬a}, the last element a and the least element ¬a. The boundary ∂C is identified with the boolean subalgebra of the distributive lattice (C, ≤_{a}) consisting of those elements which have (unique) complements.

A median set X is said to be locally boolean if any cell C of X coincides with its boundary ∂C.

One cheks easily that for finite subsets A and B of a median set X, the convex set [A] ∩ [B] is either empty or it is the finitely generated convex median subset [{∨_{a} B | a ∈ A}] = [{∨_{b} A | b ∈ B}], where ∨_{a} B denotes the join of the finite set B

under the order ≤_{a}. In particular, for a, b, c, d ∈ X, the intersection of the cells [a, b]

and [c, d] is either empty or the cell [a ∨_{c} b, a ∨_{d} b] = [c ∨_{a} d, [c ∨_{b} d].

A convex subset P of a median set X is said to be prime if its complement ¬P := X\P is convex too. Thus the set SpecX of all prime convex subsets of X is closed under the operator ¬ of taking the complement and contains the empty set ∅ as well as the whole X.

For any subset A of X, set U(A) = {P ∈ SpecX | P ∩A = ∅} and V (A) = {P ∈ SpecX | A ⊆ P }. According to [7] Theorem 5.2.1., for all A, B ⊆ X, V (A) ∩ U(B) is non-empty iff [A] ∩ [B] is empty. In particular, [A] = ∩_{P ∈V (A)} P.

The family of the subsets of SpecX having the form U(A), with A ⊆ X finite, contains SpecX = U(∅) and is closed under finite intersection, and hence it is a base of a topology on SpecX, called the spectral topology. Note that the subfamily of basic open sets U(a) = {P ∈ SpecX | a ∉ P } for a ∈ X generate the spectral topology on SpecX. For any P ∈ SpecX, the closure of {P } is V (P), i.e. the specialization relation on SpecX coincides with the inclusion. In particular, as SpecX = V (∅), SpecX is irreducible with the unique generic point ∅. On the other hand, as {X} = V (X), X is the unique closed point of SpecX. Note also that SpecX is quasi-compact since U(∅) = SpecX is the unique basic open set containing X.

According to [7] Theorem 6.4., the correspondence X ↦ SpecX extends to a duality between the category of median sets and a category of irreducible spectral (coherent) spaces with a suitable additional structure. As a consequence of the duality above, any finitely generated median set is finite, and hence any median set is a direct limit of finite median sets.

A median set X is called simplicial if any cell of X has finitely many elements. In a simplicial median set the finitely generated convex subsets are finite. In particular, any simplicial median set is the direct limit of its finite convex subsets.

A median set X is said to be locally linear if the following equivalent conditions are satisfied:

i) any cell of X has at most two ends; ii) for all x, y, z ∈ X, z ∈ [x, y] implies [x, y] = [x, z] ∪ [z, y]; iii) for any cell C of X and for any a ∈ ∂C, the restriction of the partial order

≤_{a} on C is total (linear) with the last element a and the least element ¬a;

iv) for all P, Q ∈ SpecX, either P and Q are comparable (i.e. P ⊆ Q or Q ⊆ P) or else P and ¬Q are comparable.

The cells of a locally linear median set may be also called closed segments. The usual simplicial trees (i.e. acyclic connected graphs) are naturally identified

with the locally linear simplicial median sets.

1.2 Directions and foldings on median sets. Given a median set X, a map φ : X → X is called a folding (or a retraction) if φ(Y (x, y, z)) = Y (φ(x), y, φ(z)) for all x, y, z ∈ X.

By [9] Lemma 1.5.2., a necessary and sufficient condition for a map φ : X → X to be a folding is that the following conditions are satisfied:

i) φ is an endomorphism of X, ii) φ is idempotent, i.e. φ(φ(x)) = φ for all x ∈ X, and iii) the image φ(X) of φ is a convex subset of X.

According to [9] Proposition 7.3., the map φ ↦ φ(X) maps bijectively the set F old(X) of foldings of X onto the set of those non-empty convex subsets I of X which are retractible, i.e. satisfy the following equivalent conditions:

a) there exists a (unique) median set retract p : X → I of the embedding I → X;

b) for all (for some) a ∈ I and for all b ∈ X, [a, b] ∩ I is a cell;

c) for all x ∈ X, the join ∨_{x} I with respect to the order ≤_{x} exists and

belongs to I.

Note that in a simplicial median set any non-empty convex subset is retractible. The set F old(X) of foldings of X has a canonical structure of median set with

the median Y defined point-wise : Y (φ_{1}, φ_{2}, φ_{3})(x) = Y (φ_{1}(x), φ_{2}(X), φ_{3}(x)), and the map X → F old(X) assigning to any a ∈ X the constant folding x ↦ a identifies X with a median subset of F old(X).

By a direction on X we understand a semilattice operation ∨ on X satisfying the following compatibility condition with the median operation on X : for any a ∈ X, the translation φ_{a} : X → X, x ↦ a ∨ x is a folding of X.

By [9] Proposition 3.7., the directions on X correspond bijectively to the partial orders ≤ on X satisfying

i) ≤ is compatible with Y, i.e. for all x, y, z, u ∈ X, Y (x, y, z) ≤ Y (x, y, u) whenever z ≤ u, and

ii) any pair (x, y) of elements in X is bounded above.

Any element a of X determines a direction ∨_{a} on X given by x ∨_{a} y = Y (x, a, y). Such directions are called internal or closed, while the other ones, if exist, are called external or open.

According to [9] Corollary 8.3., if D_{i}, i = 1, 2, 3, are directions on X then the binary operation ∨ on X given by a ∨ b = Y (a ∨_{D_{1}} b, a ∨_{D_{2}} b, a ∨_{D_{3}} b) defines a direction

Y (D_{1}, D_{2}, D_{3}) on X. Thus the set Dir(X) of directions on X has a canonical structure of median set, and the map X → Dir(X), a ↦ ∨_{a}, identifies X with a convex subset of Dir(X). By [9] Corollary 8.4., Dir(X) is locally linear (resp. locally boolean) iff X is locally linear (resp. locally boolean). Moreover, assuming X is non-empty, it follows by [9] Proposition 8.5. that X is existentially closed in Dir(X), i.e. Dir(X) is embeddable over X into an ultrapower of X.

By [9] Proposition 8.2., the median set Dir(X) admits a presentation as the inverse limit lim_{←} I of the non-empty finitely generated convex median subsets I of X with the connecting epimorphisms p_{I,J} : J → I for I ⊆ J given by p_{I,J}(x) = ∨_{x} I. Thus Dir(X) has a natural prodiscrete topology inducing the discrete topology on X. In particular, Dir(X) is a boolean space whenever the median set X is simplicial.

According to [9] Proposition 9.3., given a median set X, the associated median sets F old (Dir(X)) and Dir (F old (X) are canonically isomorphic over X :

φ ∈ F old (Dir (X)) ↦ D_{φ} ∈ Dir (F old (X)), where (ψ ∨_{D_{φ}} θ)(x) = ψ(x) ∨_{φ(∨_{x})} θ(x)

for ψ, θ ∈ F old (X), x ∈ X.

1.3 Λ-metric median sets. Let (Λ; +, ≤, ∧, ∨) denote a l.o. abelian group. By a Λ–metric space we understand a set X together with a Λ–valued ”distance” map d : X × X → Λ satisfying the usual axioms

i) d(x, y) = 0 iff x = y,

ii) d(x, y) = d(y, x) for all x, y ∈ X, and
iii) (the triangle inequality) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X.

As 0 = d(x, x) ≤ d(x, y) + d(y, x) = 2d(x, y) it follows that d takes values in Λ_{+} := {α ∈ Λ | α ≥ 0}.

For x, y ∈ X, set [x, y] = {z ∈ X | d(x, z) + d(z, y) = d(x, y)}.

The metric d on X induces the betweenness relation xzy ↔ z ∈ [x, y]. Note that [x, y] = [x, z] implies y = z, and z ∈ [x, y] and u ∈ [x, z] imply z ∈ [u, y].

The next lemma is immediate.

Lemma 1.1 The following are equivalent for a Λ–metric space (X, d) :

i) For all x, y, z ∈ X, the intersection [x, y] ∩ [y, z] ∩ [z, x] consists of a single element denoted by Y (x, y, z).

ii) For all x, y, z ∈ X there exists u ∈ X such that [x, y] ∩ [x, z] = [x, u] and u ∈ [y, z].

A Λ–metric space (X, d) is said to be a Λ–metric median set (or a generalized Λ–tree) if (X, d) satisfies the equivalent conditions of Lemma 1.1.

Any l.o. abelian group Λ has a canonical underlying structure of Λ–metric median set with d(α, β) = |α − β|, where |α| = α ∨ −α. For any α ∈ Λ, the translation x ↦ x + α is an automorphism of the Λ–metric median set Λ.

A Λ–metric set X is said to be (locally) full, resp. (locally) faithfully full, if for all x, y ∈ X, the map [x, y] → [0, d(x, y)], z ↦ d(x, z) is onto, resp. bijective. Note that the map above is a Λ–isometry whenever X is faithfully full.

Remark 1.2 Assuming Λ is totally ordered, the faithfully full Λ–metric median sets are exactly the Morgan–Shalen Λ–trees [19], [2].

Proposition 1.3 Given a l.o. abelian group Λ, a set X and a mapping d : X × X → Λ satisfying d(x, y) = 0 iff x = y, and d(x, y) = d(y, x), set (x, y)_{z} = \frac{1}{2}(d(x, z) + d(z, y) − d(x, y)) ∈ ℤ [\frac{1}{2}] ⊗_{ℤ} Λ for x, y, z ∈ X. A necessary and sufficient condition for d to make X a faithfully full Λ–metric median set is that the following are satisfied:

i) (x, y)_{z} ∈ Λ for all x, y, z ∈ X,

ii) (x, y)_{u} ≥ (x, z)_{u} ∧ (z, y)_{u} for all x, y, z, u ∈ X, and

iii) the map [x, y] → [0, d(x, y)], z ↦ d(x, z) is onto for all x, y ∈ X.

Proof Assuming (X, d) is a faithfully full Λ–metric median set, we get (x, y)_{z} = d(z, Y (x, y, z)) ∈ Λ, i.e. i), while iii) is obvious. To get ii) note that [u, Y (x, z, u)] ∩ [u, Y (y, z, u)] = [u, Y (Y (x, z, u), Y (y, z, u), u)] = [u, Y (Y (x, y, u), z, u)] = [u, z] ∩ [u, Y (x, y, u)]. As the canonical map [u, z] → [0, d(u, z)] is bijective by assumption, it follows that (x, y)_{u} = d(u, Y (x, y, u)) ≥ d(u, Y (Y (x, z, u), Y (y, z, u), u) = d(u, Y (x, z, u))∧d(u, Y (y, z, u) = (x, z)_{u} ∧(y, z)_{u}, as contended.

Conversely, assuming the conditions i) - iii) are satisfied, it follows by ii) that (x, y)_{z} ≥ (x, z)_{z} ∧ (y, z)_{z} = 0, whence (X, d) is a Λ–metric space, so it remains to show that the canonical map [x, y] → [0, d(x, y)] is injective (and hence bijective by iii)) for all x, y ∈ X, and the set [x, y] ∩ [y, z] ∩ [z, x] is a singleton for all x, y, z ∈ X. To get the injectivity of the map above, let z, u ∈ [x, y] be such that d(x, z) = d(x, u). By iii) it follows that 2d(x, z) − d(z, u) = 2(z, u)_{x} ≥ 2((z, y)_{x} ∧ (u, y)_{x}) = 2d(x, z), whence d(z, u) = 0, i.e. z = u.

Finally, assuming u, v ∈ [x, y] ∩ [y, z] ∩ [z, x], we get d(x, u) = d(x, v) = (y, z)_{x}, whence u = v thanks to the injectivity of the canonical map [x, y] → [0, d(x, y)]. Thus, it remains to show that the intersection above is non-empty. By i) and iii) there exists u ∈ [x, y] such that d(x, u) = (y, z)_{x}, so it suffices to show that u ∈ [x, z], i.e. d(z, u) = (x, y)_{z}. By the triangle inequality we get d(z, u) ≥ d(x, z) − d(x, u) = (x, y)_{z}. On the other hand, it follows by ii) that 0 = 2(x, y)_{u} ≥ 2((x, z)_{u}∧(y, z)_{u}) = d(z, u) − (x, y)_{z}, i.e. d(z, u) = (x, y)_{z} as contended.

1.4 Λ–valued lattices. Given an abelian group Λ and a lattice (L; ≤, ∧, ∨), by a Λ–valued distribution on L we understand a map v : L → Λ satisfying the identity v(x) + v(y) = v(x ∧ y) + v(x ∨ y) for all x, y ∈ L.

Assuming that the abelian group Λ is l.o., a Λ–valued distribution v : L → Λ is called a Λ–valuation if in addition x < y implies v(x) < v(y).

In particular, v(x ∧ y) ≤ v(x) ∧ v(y) and v(x) ∨ v(y) ≤ v(x ∨ y) for all x, y ∈ L. If v : L → Λ is a Λ–valuation then for any α ∈ Λ, the map v_{α} : L → Λ, x ↦

v(x) + α, is a Λ–valuation too.

The following two lemmata are easy.

Lemma 1.4 The lattice L is modular whenever there exists a Λ-valuation v : L → Λ on L.

A Λ–valuation v : L → Λ induces a Λ-metric d_{v} : L × L → Λ, (x, y) ↦ v(x ∨ y) − v(x ∧ y). Note that d_{v}(x, y) = d_{v}(x ∧ y, x ∨ y) ≥ |v(x) − v(y)| for all x, y ∈ L, and d_{v_{α}} = d_{v} for any α ∈ Λ. For x, y, z ∈ L, we get (x, y)_{z} = d_{v}((x ∧ y) ∨ z, (x ∧ z) ∨ (y ∧ z)) = d_{v}((x ∨ z) ∧ (y ∨ z), (x ∨ y) ∧ z).

For x, y ∈ L, set ⟨x, y⟩ := {z ∈ L | d_{v}(x, z) + d_{v}(z, y) = d_{v}(x, y)} = {z ∈ L | (x∧y)∨(y∧z)∨(z∧x) = z} = {z ∈ L | (x∨y)∧(y∨z)∧(z∨x) = z}. It follows that ⟨x, y⟩ ⊆ [x, y] := {z ∈ L | x ∧ y ≤ z ≤ x ∨ u}, i.e. the canonical betweenness relation on the lattice L is finer that the betweenness relation ⟨x, z, y⟩ ↔ z ∈ ⟨x, y⟩ induced by the Λ-metric d_{v}.

Lemma 1.5 Given a Λ-valuation v : L → Λ, the following are equivalent:

i) L is a distributive lattice.

ii) d_{v}(x, y) + d_{v}(y, z) + d_{v}(z, x) = 2d_{v}(x ∨ y ∨ z, x ∧ y ∧ z) for all x, y, z ∈ L. iii) The betweenness relations above coincide, i.e. ⟨x, y⟩ = [x, y] for all x, y ∈ L.

2 Arboreal structures on Prüfer domains

2.1 Preliminaries. Let A be a Prüfer domain and let K denote its field of fractions. By [15] Ch.4, [13] §2, the non-zero fractional ideals of finite type form an abelian group under multiplication denoted D(A), with A as neutral element and the inverse given by α^{−1} = [A : α] = {x ∈ K | xα ⊆ A}. Since sometimes it is convenient to use an additive notation, let Λ := Div(A) = {(α) | α ∈ D(A)} denote the additive group of divisors of K. W.r.t. the partial order (α) ≤ (β) iff β ⊆ α, Λ becomes a l.o. abelian group with the (distributive) lattice operations (α) ∧ (β) = (α+β) and (α)∨(β) = (α∩β). Setting 0 = (A), let Λ_{+} = {(α) | α ∈ D(A), α ⊆ A} denote the monoid of nonnegative divisors of K corresponding to non-zero finitely generated ideals of A. For any α ∈ D(A), set (α)_{+} = (α ∩ A), (α)_{−} = (−(α))_{+} = (α^{−1} ∩ A) and |(α)| = (α ∩ α^{−1}). Let P = P(A) = {(Ax) | x ∈ K^{∗}} denote the subgroup of principal divisors. By convention, set (0) = ∞ ≥ (α) for all α ∈ D(A). Note that for all x ∈ Λ there are finitely many y_{i} ∈ P, i = 1, …, n, such that x = y_{1} ∧…∧y_{n}. Notice also that Λ = P iff A is a Bézout domain. Given α ∈ D(A), consider the map | |_{α} : K → Λ_{+}, x ↦ |x|_{α} = (α(α + Ax)^{−1}) = ((α) − (x))_{+}. As x + y ∈ Ax + Ay, i.e. (x) ∧ (y) ≤ (x + y), we get |x + y|_{α} ≤ |x|_{α} + |y|_{α}. Note also that |−x|_{α} = |x|_{α}, |x|_{α} ≤ |x|_{β} whenever β ⊆ α, and |x|_{α} = 0 iff x ≡ 0 mod α. Thus the maps K × K → Λ_{+}, (x, y) ↦ |x − y|_{α}, for α ∈ D(A), are pseudometrics on K inducing Λ–metric space structures on the quotients X_{α} := K/α.

Further let V = K^{2} be the two-dimensional vector space over K and let L(A) denote the set of the lattices of V of finite type, i.e. the set of the finitely generated A-submodules of V spanning the vector space V. With respect to the opposite of inclusion, L(A) becomes a lattice with the sum L_{1} + L_{2}, resp. the intersection L_{1} ∩ L_{2}, as meet, resp. join, of any pair (L_{1}, L_{2}).

The group D(A) acts freely on the lattice L(A) : (α, L) ↦ αL. Set X = X(A) := D(A)\L(A) with the corresponding equivalence relation on L(A) : L_{1} ∼ L_{2} iff L_{1} = αL_{1} for some (for a unique) α ∈ D(A). Notice that the canonical action from the left of GL(V) on the lattice L(A) induces an action of GL(V) on X.

Proposition 2.1 Given L ∈ L(A) and 0 ≠ e ∈ V, there is a unique α ∈ D(A) such that Ke ∩ L = αe. Moreover, the finitely generated A-submodule Ke ∩ L is a direct summand of L, i.e. there are 0 ≠ f ∈ V and β ∈ D(A) such that e and f form a base of V and L = αe ⊕ βf.

Proof Let 0 ≠ g ∈ V be such that e and g form a base of V, and let x_{i} = a_{i}e + b_{i}g, i = 1, … n, be generators of the lattice L. Notice that not all a_{i}’s (resp. b_{i}’s) can be zero. Let γ ∈ D(A) be the fractional ideal generated by the b_{i}’s, i = 1, …, n. Multiplying with γ^{−1} we may assume without loss that γ = A, i.e. b_{i} ∈ A for i = 1, …, n and ∑_{i=1}^{n} b_{i}c_{i} = 1 for some c_{i} ∈ A. It follows that the vectors e and f := ∑_{i=1}^{n} c_{i}x_{i} = (∑_{i=1}^{n} c_{i}a_{i}) e + g ∈ L form a base of V and x_{i} = (a_{i} − b_{i} ∑_{j=1}^{n} c_{j}a_{j}) e + b_{i}f, i = 1, …, n, whence L = αe ⊕ Af where

α = ∑_{i=1}^{n} A (a_{i} − b_{i} ∑_{j=1}^{n} c_{j}a_{j}) ∈ D(A).

Corollary 2.2 Fix a base (e_{1}, e_{2}) of V = K^{2}.

i) The map D(A) × D(A) × K → L(A), (α, β, a) ↦ αe_{1} + β(ae_{1} + e_{2}) induces a bijection D(A)×D(A)×K/_{≡} → L(A), where (α, β, a) ≡ (α′, β′, a′) iff α = α′, β = β′ and a ≡ a′ mod αβ^{−1}. In other words, L(A) is identified with the disjoint union

∪_{(α,β)∈D(A)^{2}}^{·} X_{αβ^{−1}}, where X_{α} = K/α for α ∈ D(A).

ii)Given the lattices M = αe_{1} + β(ae_{1} + e_{2}) and N = γe_{1} + δ(be_{1} + e_{2}), the lattices M + N and M ∩ N admit the decompositions:

M + N = (α + γ + (β ∩ δ)(a − b))e_{1} ⊕ (β + δ)((ξa + (1 − ξ)b)e_{1} + e_{2}), where ξ ∈ A ∩ βδ^{−1}, 1 − ξ ∈ A ∩ β^{−1}δ, and

M ∩ N = (α ∩ γ)e_{1} ⊕ (β ∩ δ ∩ (α + γ)(a − b)^{−1})(ce_{1} + e_{2}), where c ≡ a mod α(β^{−1} + δ^{−1} + (α^{−1} ∩ γ^{−1})(a − b)), and c ≡ b mod γ(β^{−1} + δ^{−1} + (α^{−1} ∩ γ^{−1})(a − b)).

Remark 2.3 The lattice L(A) is not distributive. Indeed, let a ∈ A be such that a ≠ 0 and a is not invertible in A, and take L = Ae_{1} ⊕ Ae_{2}, M = Ae_{1} ⊕ A(a^{−2}e_{1} + e_{2}) and N = Ae_{1} ⊕ A(a^{−1}e_{1} + e_{2}). It follows that L + M ∩ N =

L ⊂_{≠} L + N = (L + M) ∩ (L + N).

Corollary 2.4 Fix a base (e_{1}, e_{2}) of V = K^{2}. The map D(A) × K → L(A), (α, a) ↦ αe_{1} ⊕A(ae_{1} +e_{2}) induces a bijection D(A)×K/_{≡} → X(A) = D(A)\L(A), where (α, a) ≡ (β, b) iff α = β and a ≡ b mod α. Thus X(A) is identified with the disjoint union ∪_{α∈D(A)}^{·} X_{α}.

2.2 Symmetric Λ–forms on the lattice L(A). Setting V = K^{2}, let

φ : V × V → ∧^{2} V ≅ K, (x, y) ↦ x ∧ y denote the canonical alternate bilinear map.

For M, N ∈ L(A) and 0 ≠ v ∈ ∧^{2} V, we get φ(M × N) = φ(N × M) = {x ∧ y | x ∈

M, y ∈ N} = αv, where α ∈ D(A) is uniquely determined by M, N and v. Thus we get a symmetric Λ–valued map ( , )_{v} : L(A) × L(A) → Λ which is up to an additive constant independent on the choice of v, i.e. (M, N)_{w} = (M, N)_{v} + (a) if w = av, a ∈ K^{∗}.

Lemma 2.5 The symmetric Λ–form ( , )_{v} above induces a Λ–valuation L(A) → Λ, M ↦ (M, M)_{v}, on the lattice L(A). In particular, the lattice L(A) is modular.

Proof First let’s show that M = N whenever M ⊆ N and (M, M)_{v} = (N, N)_{v}, i.e. φ(M × M) = φ(N × N). By Corollary 2.2. there is a base (e_{1}, e_{2}) of V such that M = αe_{1} ⊕ βe_{2} and N = γe_{1} ⊕ δ(ae_{1} + e_{2}) with α, β, γ, δ ∈ D(A) and a ∈ K. Assuming M ⊆ N we get α ⊆ γ, β ⊆ δ and aβ ⊆ γ. As φ(M × M) = αβ(e_{1} ∧ e_{2}) and φ(N × N) = γδ(e_{1} ∧ e_{2}), the assumption φ(M × M) = φ(N × N) implies α = γ, β = δ and a ∈ αβ^{−1}, whence M = N as contended.

Next we have to show that

(M, M)_{v} + (N, N)_{v} = (M + N, M + N)_{v} + (M ∩ N, M ∩ N)_{v}.

Setting M = αe_{1} ⊕ βe_{2} and N = γe_{1} ⊕ δ(ae_{1} + e_{2}) and using Corollary 2.2. again, we get

φ(M + M) = αβ(e_{1} ∧ e_{2}), φ(N × N) = γδ(e_{1} ∧ e_{2}),
φ((M + N) × (M + N)) = (α + γ + (β ∩ δ)a)(β + δ)(e_{1} ∧ e_{2})

and

φ((M ∩ N) × (M ∩ N)) = (α ∩ γ)(β ∩ δ ∩ (α + γ)a^{−1})(e_{1} ∧ e_{2}).

The required equality is an immediate consequence of the identity x + y = x ∨ y + x ∧ y valid in any l.o. abelian group.

According to 1.4., L(A) becomes a Λ–metric space with the “distance” map d : L(A) × L(A) → Λ_{+} given by d(M, N) = (M∩N, M∩N)−(M+N, M+N) = (M, M)+(N, N)−2(M+N, M+N);

the reference vector v ∈ ∧^{2} V is omitted since d doesn’t depend on the choice of v. Obviously, the distance d is stable under the actions of D(A) and GL(V) on L(A).

The distance d on L(A) can be described alternatively in terms of the canonical distances on the l.o. abelian group Λ and the quotients X_{α} = K/α for α ∈ D(A), as follows : w.r.t. a base (e_{1}, e_{2}) of V, setting M = αe_{1} ⊕β(ae_{1} +e_{2}) and N = γe_{1} +δ(be_{1} +e_{2}), we get

d(M, N) = |(α) − (γ)| + |(β) − (δ)| + 2|a − b|_{(α+γ)(β∩δ)^{−1}}.

2.3 The “index” (M : N). Given M, N ∈ L(A), let (M : N) = {a ∈ K | aN ⊆ M}. (M : N) is an A-submodule of K and (αM : βN) = α^{−1}β(M : N) for all α, β ∈ D(A). Obviously, the “index” (M : N) is stable under the action of GL(V) on L(A).

Lemma 2.6 For all M, N ∈ L(A), (M : N) ∈ D(A) and ((M : N)) = (M, M) − (M, N).

Proof W.r.t. a base (e_{1}, e_{2}) of V, let M = αe_{1} + β(ae_{1} + e_{2}) and N = γe_{1} + δ(be_{1} + e_{2}). For any c ∈ K, cN ⊆ M iff cγ ⊆ α, c(a − b)δ ⊆ α and cδ ⊆ β, whence (M : N) = αγ^{−1} ∩ βδ^{−1} ∩ αδ^{−1}(a − b)^{−1} = αβ(αδ + βγ + βδ(a − b))^{−1}, i.e. ((M : N)) = (M, M)_{e_{1}∧e_{2}} − (M, N)_{e_{1}∧e_{2}}, as contended.

Corollary 2.7 For all M ∈ L(A), (M : M) = A, i.e. ((M : M)) = 0.

Thus, for all M, N ∈ L(A), the subset {(α) | α ∈ D(A), αN ⊆ M} of the ordered set Λ is the principal filter generated by ((M : N)).

2.4 The Λ-metric median set structure on X(A) = D(A)\L(A). Define the map d̃: L(A) × L(A) → Λ by

d̃(M, N) = ((M : N)) + ((N : M)) = (M, M) + (N, N) − 2(M, N).

Obviously, d̃(M, N) = d̃(N, M) and d̃(αM, βN) = d̃(M, N) for all α, β ∈ D(A), so d̃ induces a symmetric map d̃: X(A) × X(A) → Λ.

For any M ∈ L(A), let M̃ denote the class of M modulo D(A).

Lemma 2.8 With the notations above, d̃(M̃, Ñ) = 0 iff M̃ = Ñ.

Proof The equality d̃(M̃, M̃) = 0 is immediate by Corollary 2.7. Conversely, assuming d̃(M̃, Ñ) = 0, i.e. (M : N)(N : M) = A, it follows that N = (N : M)(M : N)N ⊆ (N : M)M ⊆ N, whence N = (N : M)M, i.e. M̃ = Ñ as contended.

Proposition 2.9 The map d̃ : X(A) × X(A) → Λ, as defined above, makes X(A) = D(A)\L(A) a faithfully full Λ–metric median set.

Proof We have to verify the conditions i)–iii) of Proposition 1.3.

For M, N, L ∈ L(A), we get (M, N)_{L} := \frac{1}{2}(d̃(M, L) + d̃(N, L) − d̃(M, N)) = (L, L) + (M, N) − (M, L) − (N, L) ∈ Λ, as required.

To get the inequality (M, N)_{L} ≥ (M, T)_{L} ∧ (N, T)_{L} for all M, N, T, L ∈ L(A), we have to show that (M, N) + (T, L) ≥ ((M, T) + (N, L)) ∧ ((M, L) + (N, T)). Fix a base (e_{1}, e_{2}) of V = K^{2} and let m = m_{1}e_{1} + m_{2}e_{2} ∈ M, n = n_{1}e_{1} + n_{2}e_{2} ∈ N, t = t_{1}e_{1} + t_{2}e_{2} ∈ T, l = l_{1}e_{1} + l_{2}e_{2} ∈ L, Δ_{m,n} = det(m, n) = m_{1}n_{2} − m_{2}n_{1}, Δ_{t,l}, etc. We get m ∧ n = Δ_{m,n}(e_{1} ∧ e_{2}), t ∧ l = Δ_{t,l}(e_{1} ∧ e_{2}), etc. The inequality above is a consequence of the identity Δ_{m,n}Δ_{t,l} = Δ_{m,t}Δ_{n,l} − Δ_{n,t}Δ_{m,l}.

To conclude that (X(A), d̃) is a faithfully full Λ-metric median set it remains to check that the map {L ∈ L(A) | d̃(M, L) + d̃(L, N) = d̃(M, N)} → [0, d̃(M, N)], L ↦ d̃(M, L), is onto for all M, N ∈ L(A). We may assume without loss that N ⊆ M and (M : N) = A, whence d̃(M, N) = ((N : M)), since otherwise we can replace N with (M : N)N. Let α ∈ D(A) be such that (N : M) ⊆ α, and let L = N +αM. We have to show that d̃(L, M) = (α) and d̃(L, N) = ((N : M))−(α). First note that (M : L) = (L : N) = A. Indeed, A ⊆ (M : L) since L ⊆ M, and (M : L)N ⊆ (M : L)L ⊆ M, as N ⊆ L, whence (M : L) ⊆ (M : N) = A. Further A ⊆ (L : N) as N ⊆ L, and, on the other hand, (L : N)N ⊆ L ⊆ M, whence (L : N) ⊆ (M : N) = A.

We get : d̃(L, M) = ((L : M)) = (L, L) − (L, M) = (N + αM, N + αM) − (N + αM, M) = ((N, N) ∧ ((M, N) + (α)) ∧ ((M, M) + 2(α))) − ((M, N) ∧ ((M, M) + (α))) = (α) since (N, N) = ((N : M)) + (M, N) ≥ (α) + (M, N).

On the other hand, d̃(L, N) = ((N : L)) = (N, N) − (N, L) = (N, N) − (N, N + αM) = (N, N) − (N, N) ∧ ((N, M) + (α)) = (((N : M)) − (α))_{+} = ((N : M)) − (α), as contended.

Corollary 2.10 For M, N, L ∈ L(A), the median Y (M̃, Ñ, L̃) of the triple (M̃, Ñ, L̃) is the class mod D(A) of the lattice T := (L : M)M + (L : N)N.

Proof Let α ∈ D(A) be such that (α) = (L, N)_{M}. According to the last part of the proof of Proposition 2.9., Y (M̃, Ñ, L̃) is the class mod D(A) of the lattice T′ := αM + (M : N)N. One checks easily that T̃′ = T̃, as contended.

Remark 2.11 For M, N ∈ L(A), d(M, N) − d̃(M, N) =

2((M, N) − (M + N, M + N)) = 2((M, N) − (M, M) ∧ (N, N) ∧ (M, N)) = 2(((M : N))_{−} ∨ ((N : M)))_{−} ≥ 0.

Thus d(M, N) = d̃(M, N) iff (M : N) + (N : M) ⊆ A.

Fix a base (e_{1}, e_{2}) of V = K^{2}. By Corollary 2.4., the map (α, a) ↦ αe_{1} + A(ae_{1} + e_{2}) induces a bijection of the disjoint union ∪_{α∈D(A)} X_{α} onto X(A), where

X_{α} = K/α for α ∈ D(A). The distance d̃ on X(A) can be described in terms of the canonical distances on Λ and the X_{α}’s as follows: for M = αe_{1} + A(ae_{1} + e_{2}) and N = βe_{1} + A(be_{1} + e_{2}),

d̃(M̃, Ñ) = d(M, N) = |(α) − (β)| + 2|a − b|_{α+β}.

Thus the canonical projection L(A) → X(A) admits a section identifying X(A) with a Λ–metric subspace of L(A), X_{e_{1},e_{2}} = {αe_{1}+A(ae_{1}+e_{2}) | α ∈ D(A), a ∈ X_{α}}. Note that X_{e_{1},e_{2}} is a Λ-metric median set of representatives of L(A) mod D(A), however X_{e_{1},e_{2}} is not convex in the Λ-metric space L(A). Indeed, let M = Ae_{1} + Ae_{2}, N = Ae_{1} + A(a^{−1}e_{1} + e_{2}) ∈ X_{e_{1},e_{2}}, where a ∈ A\A^{∗}, and let L = Ae_{1} + Aae_{2}. We get M ∩N = L and M +N = Aa^{−1}e_{1}+A(ae_{1}+e_{2}), whence d(M, L) = d(N, L) = (a) and d(M, N) = d̃(M, N) = 2(a) = d(M, L) + d(L, N), however L ∉ X_{e_{1},e_{2}}.

Note that X_{e_{1},e_{2}} is closed under the sum: M + N ∈ X_{e_{1},e_{2}} whenever M and N belong to X_{e_{1},e_{2}}. Moreover, the semilattice operation + is compatible with the median operation Y, inducing an external direction D_{e_{1},e_{2}} on the median set X(A). Indeed, let’s show that Y (M, N, L) ⊆ Y (M, N, T) for all M, N, L, T ∈ X_{e_{1},e_{2}} subject to L ⊆ T. As Y (M, N, L) = ((L : M) + (L : N))^{−1}((L : M)M + (L : N)N), we have to show that (L : M)((L : M) + (L : N))^{−1} ⊆ (T : M)((T : M) + (T : N))^{−1}, i.e. (T, M) + ((L, M) ∨ (L, N)) ≥ (L, M) + ((T, M) ∨ (T, N)). Setting M = αe_{1} + A(ae_{1} + e_{2}), N = βe_{1} + A(be_{1} + e_{2}), L = γe_{1} + A(ce_{1} + e_{2}) and T = δe_{1} + A(ce_{1} + e_{2}) with γ ⊆ δ, we get (T, M) + ((L, M) ∨ (L, N)) = (α + δ + A(a − c))(γ + (α + A(a − c)) ∩ (β + A(b − c)))(α + γ + A(a − c)(δ + (α + A(a − c)) ∩ (β + A(b − c))) = (L, M) + ((T, M) ∨ (T, N)) as required. As for any M ∈ X_{e_{1},e_{2}}, [M, D_{e_{1},e_{2}}) = {N ∈ X_{e_{1},e_{2}} | M ⊆ N} is infinite, the direction D_{e_{1},e_{2}} is external. GL(V) acts transitively on the set of external directions {D_{e_{1},e_{2}} | (e_{1}, e_{2}) base of V } : σD_{e_{1}, e_{2}} = D_{σ(e_{1}), σ(e_{2})}.

2.5 X(A) as a collection of affine subsets of K. A subset M of the field K is called A–affine if for all natural numbers n, x_{i} ∈ M, a_{i} ∈ A for

i = 1, …, n, ∑_{i=1}^{n} a_{i}x_{i} ∈ M whenever ∑_{i=1}^{n} a_{i} = 1.

The next lemma is immediate.

Lemma 2.12 The map (α, x mod α) ↦ x + α = {x + y | y ∈ α} maps bijectively the set of pairs (α, x mod α), where α is an A–submodule of K and x ∈ K/α, onto the set of non-empty A–affine subsets of K.

Fix a base (e_{1}, e_{2}) of V = K^{2}. By Corollary 2.4. and Lemma 2.12., X(A) = D(A)\L(A) is identified with the set of those non-empty A–affine subsets M of K whose associated A–module α(M) belongs to D(A). The directed median set structure on X(A) as described in 2.4. admits an alternative description in terms of A–affine sets as follows. If M_{i} = x_{i} + α_{i} ∈ X(A), i = 1, 2, then M_{1} + M_{2} := x_{1} + (α_{1} + α_{2} + A(x_{1} − x_{2})) ∈ X(A) is the A–affine subset of K generated by the union M_{1} ∪ M_{2}, while for M_{i} = x_{i} + α_{i} ∈ X(A), i = 1, 2, 3, Y (M_{1}, M_{2}, M_{3}) = (M_{1} +M_{2})∩(M_{2} +M_{3})∩(M_{3} +M_{1}) ∈ X(A). Indeed, as A is a Prüfer domain there is x ∈ K such that x ≡ x_{i} ≡ x_{j} mod α_{i} +α_{j} +A(x_{i} −x_{j}) for 1 ≤ i < j ≤ 3, whence

Y (M_{1}, M_{2}, M_{3}) = x+α, where α = ∩_{1≤i<j≤3}(α_{i} +α_{j} +A(x_{i} −x_{j})) = Y (α_{1}, α_{2}, α_{3})+

α_{1}∩A(x_{2}−x_{3})+α_{2}∩A(x_{3}−x_{1})+α_{3}∩A(x_{1}−x_{2})+A(x_{1}−x_{2})∩A(x_{2}−x_{3})∩A(x_{3}−x_{1}). The following two lemmata are easy.

Lemma 2.13 Let M_{i} = x_{i} + α_{i} ∈ X(A), i = 1, 2, 3.

i) If M_{1} ∩ M_{2} is non-empty (and hence M_{1} ∩ M_{2} ∈ X(A)) then

(M_{1} ∩ M_{2}) + M_{3} = (M_{1} + M_{3}) ∩ (M_{2} + M_{3})

ii) If M_{1} ∩ M_{3} and M_{2} ∩ M_{3} are non-empty then

(M_{1} + M_{2}) ∩ M_{3} = (M_{1} ∩ M_{3}) + (M_{2} ∩ M_{3})

Lemma 2.14 Let n ≥ 1, M_{i} = x_{i} + α_{i} ∈ X(A), i = 1, …, n, I = [M_{1}, …, M_{n}] the convex subset of X(A) generated by the finite set {M_{1}, …, M_{n}}, and φ_{I} its associated folding. For all M = x + α ∈ X(A),

φ_{I}(M) := ∨_{M} I = ∨_{M}{M_{1}, …, M_{n}} = ( ∩_{i=1}^{n}(M_{i} + M)) ∩ (∑_{i=1}^{n} M_{i}) =

y + ( ∩_{i=1}^{n}(α + α_{i} + A(x − x_{i}))) ∩ (∑_{i=1}^{n} α_{i} + ∑_{i=2}^{n} A(x_{1} − x_{i})), where

y ≡ x mod α + α_{i} + A(x − x_{i}) for i = 1, …, n, and

y ≡ x_{1} mod ∑_{i=1}^{n} α_{i} + ∑_{i=2}^{n} A(x_{1} − x_{i}).

2.6 The action of GL(V) on the Λ–metric median set X(A). Fix a base (e_{1}, e_{2}) of V = K^{2} and identify X(A) with the disjoint union ∪_{α∈D(A)}^{·} X_{α},

where X_{α} = K/α. For σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(K) and λ mod α ∈ X(A), setting

Δ = det(σ) and γ = αc + A(λc + d), we get σ(λ mod α) = (ϵa + ρ(λa + b)) mod αγ^{−2}Δ, where ϵ ∈ γ^{−1}α and ρ ∈ γ^{−1} such that ϵc + ρ(λc + d) = 1. In particular, for c = 0, i.e. for σ belonging to the Borel subgroup B of GL_{2}(K), we get σ(λ mod α) = d^{−1}(λa + b) mod αad^{−1}. Thus the action of B on X(A) is induced by the action of B on D(A), resp. K, given by σ · α = ad^{−1}α, resp. σ · λ = d^{−1}(λa + b).

One checks easily that the kernel of the action of GL_{2}(K) on X(A) is the normal

subgroup \{( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) | a ∈ K^{∗}\} ≅ K^{∗}, inducing a faithfull action of P GL_{2}(K) =

GL_{2}(K)/K^{∗} on X(A).

For any α ∈ D(A), the subgroup P = \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a ∈ K^{∗}, b ∈ K\} ≅ K ⋊ K^{∗}

of B acts transitively on the subset ∪_{a∈K^{∗}/A^{∗}}^{·} X_{αa}, and the stabilizer of (α, 0) in P

is \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a ∈ A^{∗}, b ∈ α\} ≅ α ⋊ A^{∗}. The surjective map X(A) → Λ,

λ mod α ↦ (α) induces the bijections X(A)\P → X(A)\B → Λ/P =: C, where C = Λ/P is the group of divisor classes of K.

Lemma 2.15 Assume that the Prüfer domain A satisfies the following condition : for all α, β ∈ D(A) such that α ⊆ β, the A-modules β/α and A/αβ^{−1} are isomorphic, i.e. there exists x ∈ K such that β = α + Ax (in particular, for any α ∈ D(A) there are x, y ∈ K such that α = Ax + Ay; note that the condition above is satisfied by Dedekind domains). The surjective map X(A) → Λ, λ mod α ↦ (α) induces the bijections GL_{2}(K)\X(A) → Λ/2Λ + P ≅ C/2C and SL_{2}(K)\X(A) → Λ/2Λ.

Proof The canonical mappings above are onto for any Prüfer domain A. To get their injectivity, for arbitrary α, γ ∈ D(A), λ ∈ K and Δ ∈ K^{∗}, let 0 ≠ c ∈ α^{−1}γ. As αc ⊆ γ there is d ∈ γ such that γ = αc + Ad, i.e. αγ^{−1}c + γ^{−1}d = A. Choose ϵ ∈ αγ^{−1} and ρ ∈ γ^{−1} such that ϵc + ρd = 1, and consider the linear system in two variables a and b :

\{ \begin{matrix}ϵa + ρb = λ \\ da − cb = Δ. \end{matrix}

As its determinant is −ϵc − ρd = −1 ≠ 0, the linear system above has by

Cramer’s rule a unique solution (a, b). Thus σ := ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(K) and

σ(0 mod α) = λ mod αγ^{−2}Δ as required.

Now let’s evaluate the distance d̃(x, σ(x)) for x ∈ X(A), σ ∈ GL_{2}(K).

Lemma 2.16 Let λ mod α ∈ X(A) and σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(K). Set

Δ = det(σ), γ = αc + A(λc + d) and δ = A(a − λc) + α^{−1}(λa + b − λ(λc + d)). Then

d̃(λ mod α, σ(λ mod α)) = (Δ) − 2(γ + δ).

Proof Let ϵ ∈ γ^{−1}α and ρ ∈ γ^{−1} be such that ϵc + ρ(λc + d) = 1. As σ(λ mod α) = (ϵa + ρ(λa + b)) mod αγ^{−2}Δ, we get d̃(λ mod α, σ(λ mod α)) = (Δ) − 2(γ + γ^{−1}Δ + α^{−1}γ(ϵa + ρ(λa + b) − λ)), so it suffices to check that δ′ := γ^{−1}Δ + α^{−1}γ(ϵa + ρ(λa + b) − λ) = δ.

As Δ = ad − bc = (a − λc)(λc + d) − (λa + b − λ(λc + d)) ∈ γδ, it follows that γ^{−1}Δ ⊆ δ. Since ϵa + ρ(λa + b) − λ = ϵa + ρ(λa + b) − λ(ϵc + ρ(λc + d)) = ϵ(a − λc) + ρ(λa + b − λ(λc + d)) ∈ γ^{−1}αδ, we get δ′ ⊆ δ.

On the other hand, a−λc = a(ϵc+ρ(λc+d))−λc = c(ϵa+ρ(λa+b)−λ)+ρΔ ∈ δ′, and λa + b − λ(λc + d) = (λa + b)(ϵc + ρ(λc + d)) − λ(λc + d) = (λc + d)(ϵa + ρ(λa + b) − λ) − ϵΔ ∈ αδ′, and hence δ ⊆ δ′, as contended.

Corollary 2.17 d̃(0 mod α, σ(0 mod α)) = (Δ) − 2(Aa + α^{−1}b + αc + Ad). In particular, d̃(0 mod A, σ(0 mod A)) = (Δ) − 2(Aa + Ab + Ac + Ad).

Proposition 2.18 The stabilizer in GL_{2}(K) of the direction + on X(A) is the Borel subgroup B.

Proof For λ mod α, μ mod β ∈ X(A) and σ = ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) ∈ B, we get

σ(λ mod α) + σ(μ mod β) = d^{−1}(λa + b) mod αad^{−1} + d^{−1}(μa + b) mod βad^{−1} = d^{−1}(λa + b) mod (α + β + A(λ − μ))ad^{−1} = σ(λ mod (α + β + A(λ − μ))) = σ(λ mod α + μ mod β), whence B is contained in the stabilizer in GL_{2}(K) of the direction +.

Finally, using the Bruhat decomposition of GL_{2}(K) = B ∪^{·} UwB, where U =

\{( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) | a ∈ K\} ≅ K and w = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), it suffices to show that for all

a ∈ K, σ = ( \begin{matrix}a & −1 \\ 1 & 0 \end{matrix} ) ∈ Uw does not stabilize the direction +. Indeed, for

any γ ∈ D(A) such that γ ⊂_{≠} A, we get σ(0 mod A + 0 mod γ) = σ(0 mod A) =

a mod A ⊂_{≠} a mod γ^{−1} = σ(0 mod γ) = σ(0 mod A) + σ(0 mod γ).

2.7 Some remarkable foldings on X(A). For any δ ∈ D(A)_{+}, let’s consider the map Ψ_{δ} : X(A) → X(A) given by Ψ_{δ}(λ mod α) = λ mod (α + λδ)).

Proposition 2.19 The map Ψ_{δ} as defined above is a folding of the median set X(A) which preserves the direction + on X(A). The associated directed retractible convex subset Ψ_{δ}(X(A)) consists of those λ mod α for which λδ ⊆ α. In particular, for δ = A, Ψ_{A}(X(A)) is canonically isomorphic with the underlying median set of the l.o. abelian group Λ directed by the opposite of the canonical order.

The stabilizer in GL_{2}(K) of Ψ_{δ} is the subgroup D ∪ wD, where

D = \{( \begin{matrix}a & 0 \\ o & d \end{matrix} ) | a, d ∈ K^{∗} \} ≅ K^{∗} × K^{∗} and w = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), which doesn’t

depend on δ. The subgroup above consists of those σ ∈ GL_{2}(K) for which σ(λ mod α) = μ mod β with β(β + μA)^{−1} = α(α + λA)^{−1} for all λ mod α ∈ X(A).

Proof One checks easily that Ψ_{δ}(X(A)) = {λ mod α | α ∈ D(A), λδ ⊆ α} is a convex subset of X(A) and [λ mod α, 0 mod A] ∩ Ψ_{δ}(X(A)) = [λ mod (α + λδ), 0 mod A] for all (λ mod α ∈ X(A), whence Ψ_{δ} is a folding by [9] Proposition 7.3.

Obviously, Ψ_{δ}(λ mod α + μ mod β) = Ψ_{δ}(λ mod (α + β + A(λ − μ)) = λ mod (α + β + A(λ − μ) + λδ) = Ψ_{δ}(λ mod α) + Ψ_{δ}(μ mod β), and hence the folding Ψ_{δ} preserves the direction +.

Since, by definition, (σΨ_{δ})(x) = σ · Ψ_{δ}(σ^{−1} · x) for all σ ∈ GL_{2}(K), and Ψ_{δ} is a folding, it follows that the stabilizer of Ψ_{δ} equals the stabilizer of the retractible convex subset Ψ_{δ}(X(A)) of X(A). One checks easily that σΨ_{δ}(X(A)) = Ψ_{δ}(X(A))

whenever σ ∈ D ∪ wD. Conversely, let σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(K) be such that

σΨ_{δ}(X(A)) = Ψ_{δ}(X(A)). We distinguish the following cases :

Case 1) σ ∈ B, i.e. c = 0: Assuming σ ∉ D, i.e. b ≠ 0, let α ∈ D(A) be such that a^{−1}b ∉ αδ^{−1}. We get σ(0 mod α) = d^{−1}b mod αad^{−1} ∉ Ψ_{δ}(X(A)), a contradiction.

Case 2) ac ≠ 0: For all α ∈ D(A) subject to c^{−1}d ∈ αδ ⊈ Aa^{−1}c^{−1}Δ, we get σ(0 mod α) = c^{−1}a mod α^{−1}c^{−2}Δ ∉ Ψ_{δ}(X(A)), a contradiction.

Case 3) cd ≠ 0, a = 0: Thus Δ = −bc ≠ 0 and, for all α ∈ D(A) satisfying α ⊂_{≠} c^{−1}dδ, σ(0 mod α) = d^{−1}b mod αbcd^{−2} ∉ Ψ_{δ}(X(A)), again a contradiction.

Consequently, either b = c = 0 or a = d = 0, i.e. σ ∈ D∪wD, as contended.

Next let’s consider for any δ ∈ D(A) the map φ_{δ} : X(A) → X(A) given by φ_{δ}(λ mod α) = λ mod (α + δ).

Proposition 2.20 i) The map φ_{δ} as defined above is a folding of X(A) which preserves the direction +.

ii) For all λ mod α ∈ X(A), d̃(λ mod α, φ_{δ}(λ mod α)) = (αδ^{−1})_{+}.

iii) The stabilizer in GL_{2}(K) of φ_{δ} is the subgroup S := \{( \begin{matrix}a & b \\ 0 & d \end{matrix} ) ∈

B | ad^{−1} ∈ A^{∗} }. It consists of those σ ∈ GL_{2}(K) for which σ(λ mod α) = μ mod β with β = α for all λ mod α ∈ X(A).

iv) Let T_{δ} = \{( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) | b ∈ δ \} ≅ (δ, +). The point-wise stabilizer in GL_{2}(K)

of φ_{δ}(X(A)) is the subgroup S_{δ} = \{( \begin{matrix}a & ab \\ 0 & a \end{matrix} ) | a ∈ K^{∗}, b ∈ δ \} = K^{∗} · T_{δ}.

v) The map X(A) → φ_{δ}(X(A)) × Λ_{+}, x ↦ (φ_{δ}(x), d̃(x, φ_{δ}(x))), induces an embedding T_{δ}\X(A) ≅ S_{δ}\X(A) → φ_{δ}(X(A)) × Λ_{+}.

Proof i) To conclude that φ_{δ} is a folding we have to show by [9] Proposition 7.3. that φ_{δ}(X(A)) = {λ mod α) | δ ⊆ α} is a convex subset of X(A) and [λ mod α, 0 mod δ] ∩ φ_{δ}(X(A)) = [λ mod (α + δ), 0 mod δ] for all λ mod α ∈ X(A). Let λ mod α, μ mod β ∈ φ_{δ}(X(A)) and ρ mod γ ∈ X(A) be such that ρ mod γ ∈ [λ mod α, μ mod β]. Since δ ⊆ α ∩ β it follows that δ ⊆ Y (α, β, γ) ⊆ γ, whence ρ mod γ ∈ φ_{δ}(X(A)), as required. The equality above follows easily by simple computations. The identity φ_{δ}(x + y) = φ_{δ}(x) + φ_{δ}(y) for all x, y ∈ X(A) is immediate.

ii) is obvious.

iii) Given σ = ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) ∈ B, since σ(λ mod α) = d^{−1}(λa + b) mod αad^{−1},

it follows that σφ_{δ}(X(A)) = φ_{δ}(X(A)) (i.e. σφ_{δ} = φ_{δ}) iff ad^{−1} ∈ A^{∗}, i.e. σ ∈ S.

Assuming σ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(K)\B, i.e. c ≠ 0, let α ∈ D(A) be such that

δ + Ac^{−1}d ⊆ α ⊈ c^{−2}Δδ^{−1}. It follows that 0 mod α ∈ φ_{δ}(X(A)) and σ(0 mod α) = c^{−1}a mod α^{−1}c^{−2}Δ ∉ φ_{δ}(X(A)), whence σφ_{δ} ≠ φ_{δ}, as required.

iv) Obviously, σx = x for all σ ∈ S_{δ} and x ∈ φ_{δ}(X(A)). Assuming σx = x for

all x ∈ φ_{δ}(X(A)), it follows that σ ∈ S has the form ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) with ad^{−1} ∈ A^{∗}. As

0 mod δ = σ(0 mod δ) = d^{−1}b mod δ, we get d^{−1}b ∈ δ. Assuming a ≠ d let λ ∈ K be such that λ(1 − ad^{−1}) ∉ δ. On the other hand, we get λ mod δ = σ(λ mod δ) = d^{−1}(λa + b) mod δ = λad^{−1} mod δ, a contradiction. Thus σ ∈ S_{δ} as contended.

v) Note that S_{δ} = {σ ∈ GL_{2}(K) | φ_{δ}(σx) = φ_{δ}(x) and d̃(σx, φ_{δ}(x)) = d̃(x, φ_{δ}(x)) for all x ∈ X(A)}. Indeed, for any σ belongig to the right hand group and for any x = φ_{δ}(x) ∈ φ_{δ}(X(A)) we get d̃(σx, x) = d̃(σx, φ_{δ}(x)) = d̃(x, φ_{δ}(x)) = d̃(x, x) = 0, i.e. σx = x, whence σ ∈ S_{δ} by iv). Conversely, if σ ∈ S_{δ} ⊆ S and x ∈ X(A), then φ_{δ}(σx) = σφ_{δ}(x) = φ_{δ}(x) and d̃(σx, φ_{δ}(x)) = d̃(σx, σφ_{δ}(x)) = d̃(x, φ_{δ}(x)), as required.

Consequently, the map X(A) → φ_{δ}(X(A)) × Λ_{+}, x ↦ (φ_{δ}(x), d̃(x, φ_{δ}(x))) induces a map T \X(A) → S_{δ}\X(A) → φ_{δ}(X(A)) × Λ_{+}, so it remains to check its injectivity. Assuming that λ mod α and μ mod β ∈ X(A) have the same image through the map above, we get α+δ = β +δ, λ ≡ μ mod (α+δ), and α∩δ = β ∩δ, whence α = β and μ mod β = μ mod α belongs to the T_{δ}-orbit of λ mod α, as contended.

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