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Projective Valuations, D-Relations, and Ends of Λ-TreesI. M. Chiswell a b & E. Koudela aa School of Mathematical Sciences, Queen Mary, University of London, Mile End Road,London, UKb School of Mathematical Sciences, Queen Mary, University of London, Mile End Road,London, E1 4NS, UKVersion of record first published: 01 Feb 2007.

To cite this article: I. M. Chiswell & E. Koudela (2003): Projective Valuations, D-Relations, and Ends of Λ-Trees,Communications in Algebra, 31:9, 4547-4569

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Projective Valuations, D-Relations, andEnds of L-Trees

I. M. Chiswell* and E. Koudela

School of Mathematical Sciences, Queen Mary,University of London, Mile End Road,

London, UK

ABSTRACT

A projective valuation on a set E is a mapping w :E 4!L[f±1g,where L is an ordered abelian group, satisfying certain axioms. AD-relation on E is a four-place relation on E, again with certain prop-erties. There is a projective valuation on the set of ends of a L-tree(and on any subset, by restriction) and we show, using a constructionsuggested by Tits in the case L¼R, that every projective valuationarises in this way. Every projective valuation w defines a D-relation,and there is a simple geometric interpretation of the D-relation, givena L-tree defining w. Our main result is a converse, that any D-relationcan be defined by a projective valuation, hence arises from an embed-ding into the set of ends of a L-tree.

*Correspondence: I. M. Chiswell, School of Mathematical Sciences, QueenMary, University of London, Mile End Road, London E1 4NS, UK; E-mail:I.M.Chiswell@qmul.ac.uk.

COMMUNICATIONS IN ALGEBRA�

Vol. 31, No. 9, pp. 4547–4569, 2003

4547

DOI: 10.1081/AGB-120022809 0092-7872 (Print); 1532-4125 (Online)

Copyright # 2003 by Marcel Dekker, Inc. www.dekker.com

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1. INTRODUCTION

We assume familiarity with some basic ideas in the theory of L-trees,where L is an ordered abelian group. These are discussed in Alperin andBass (1987). Recall that a L-tree is a generalisation of a tree together withthe metric it induces on its set of vertices. The importance of L-trees isby now well-established (see [18] for a survey of their uses). There are,however, several other notions of generalised tree, which have been usedfor various purposes (see, for example, Adeleke and Neumann, 1998;Bowditch, 1999, [12] and [13]). They are defined without using a metric.A theme of the first author’s work (Chiswell, 1994, 1998) is to show that,nevertheless, some of these notions can be described by means of L-trees.

Continuing this theme, we show how another notion of generalisedtree, the ‘‘D-relation’’ of Adeleke and Neumann (1998), is related tothe idea of ‘‘projective valuation’’, due to Tits ([19]). A projective valua-tion on a set E is a mapping w :E 4!L[f±1g, where L is an orderedabelian group, satisfying certain axioms listed below. A D-relation onE is a four-place relation on E, again satisfying certain axioms givenbelow. One can define a projective valuation on the set of ends of a L-tree(Bennett, 1990, [19]), and every projective valuation arises by restrictionfrom an embedding into the set of ends of a L-tree. Every projectivevaluation w defines a D-relation in a simple manner (see Lemma 1.3),and there is a simple geometric interpretation of the D-relation in termsof a L-tree defing w. Our main result is a converse, that any D-relationcan be defined by a projective valuation, so arises from an embeddinginto the ends of a L-tree.

The projective valuation w on the ends of a L-tree is illustrated in thefollowing picture (representing the linear subtrees (a, b), (c, d) etc., asdefined in Bass (1991, Lemma 1.5). The arrow represents signed distancein the direction from a to b.

(We shall make the definition of w precise and prove in detail that theaxioms are satisfied in Sec. 2.)

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In this section we shall give the main definitions and prove that aprojective valuation defines a D-relation. In Sec. 2 we establish the con-nection between projective valuations and L-trees. A complete proofthat a projective valuation arises from an embedding into the ends ofa L-tree is given, using a construction suggested by Tits [19]. (A differ-ent proof, using a construction in the proof of Corollary E1.8 in Bass,1991, is given in Bennett, 1990. Further commentary on this proof isgiven in [15].)

An affine building of dimension 1 is an R-tree without endpointswhose apartments are lines joining pairs of ends of the tree. Tits gavehis construction to show that the set of ends of the apartments,together with the projective valuation defined above, determine theR-tree and the apartments. This was as part of a classification of affinebuildings. There is a generalisation to ‘‘affine L-buildings by Bennett(1990).

Finally, Sec. 3 is devoted to the proof of the main result, which isTheorem 3.8. We begin with the axioms for a projective valuation.

Definition 1.1. A projective valuation w on a set E 6¼ ; is a mapping withdomain the 4-tuples of pairwise distinct elements of E and range L, anordered abelian group, that satisfies the following conditions:

(PV1) w(a, b; c, d)¼w(c, d; a, b)¼�w(a, b; d, c).

(PV2) If w(a, b; c, d)¼ k > 0, then w(a, d; c, b)¼ k and w(a, c; b, d)¼ 0.

(PV3) If a, b, c, d, e2E are pairwise distinct, then w(a, b; d, e)þw(b, c; d, e)¼w(a, c; d, e).

It is convenient to extend the definition of a projective valuation w toall quadruples of elements of E, guided by the picture above, allowingone or more of the ends to coincide. Choose two elements 1, �1 notin L, with the usual convention that l < 1 and �1 < l for all l2L.Assume a 6¼ c and b 6¼ c. We define w(a, c; b, c)¼1, and extend this topermutations of a, c, b, c using PV1 and PV2, for example, w(a, b; c, c)¼0 and w(a, c; c, b)¼�1. Also if any three of a, b, c, d are equal weput w(a, b; c, d)¼ 0. With these extended definitions, the axiomsPV1–PV3 remain valid, where now in PV1 and PV2, a, b, c, d can beany quadruple of elements of E.

In order to define a ‘‘D-relation’’, it is convenient to use some of thenotation of propositional logic.

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Definition 1.2. Given a set E and a 4-place relation D on E, we call (E, D)an abstract direction set (or a D-set) if it satisfies the following conditions:

(D1) D(a, b; c, d))D(b, a; c, d)^D(a, b; d, c)^D(c, d; a, b).

(D2) D(a, b; c, d)):D(a, c; b, d).

(D3) D(a, b; c, d))D(a, b; c, e)_D(e, b; c, d).

(D4) If a 6¼ c and b 6¼ c, then D(a, b; c, c).

Note that, by D2, D(a, c; c, c) is false for all a, c2E, so by D1, if anythree of a, b, c, d are equal, then :D(a, b; c, d). We call a relation satisfy-ing these axioms a D-relation.

The picture on p. 85 of Adeleke and Neumann (1998) (reinterpreting‘‘directions in a B-set’’ to mean ‘‘ends of a L-tree’’) suggests that a D-rela-tion can be defined on the set of ends of a L-tree, by: D(a, b; c, d) is trueif and only if [a, b]\ [c, d]¼;, where [a, b]¼ (a, b)[fa, bg (meaning fagif a¼ b). If w is the projective valuation on the ends of the tree, itfollows from Lemma 2.1(3) below that D(a, b; c, d) is true if and onlyif w(a, c; b, d) > 0, hence D is a D-relation by the next lemma, whichshows that a projective valuation defines a D-relation.

Lemma 1.3. Given a projective valuation w on E, define a four-point rela-tion D by D(a, b; c, d) if and only if w(a, c; b, d) > 0. Then (E, D) is anabstract direction set.

Proof. Assume w(a, c; b, d) > 0. Then by PV2 w(a, d; b, c) > 0, that isD(a, b; d, c) is true. Similary by PV1 w(b, d; a, c) > 0 and by PV2 we havew(b, c; a, d) > 0)D(b, a; c, d). Now also by PV1 we get the following:

wða; c; b; dÞ > 0 ) wðc; a; d; bÞ > 0 ) Dðc; d; a; bÞ;

which gives D1.Using PV2 we get w(a, c; b, d) > 0)w(a, b; c, d)¼ 0):D(a, c; b, d)

which gives D2.For D3 assume w(a, c; b, d) > 0 and w(a, c; b, e)� 0. Then d 6¼ e and

from the way w is defined, no three of a, b, c, d, e are equal and the onlypossible coincidences for pairs of these elements are a¼ e and b¼ e. Butthen w(e, c; b, d)¼1, so D(e, b; c, d) holds. Thus we may assume they arepairwise distinct. Then by PV2, w(a, d; b, c) > 0 and w(a, e; b, c)� 0. ByPV3, w(a, d; b, c)þw(d, e; b, c)¼w(a, e; b, c), so w(d, e; b, c) < 0. By PV1,w(e, d; b, c) > 0, hence w(e, c; b, d) > 0, so D(e, b; c, d) holds.

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Finally, assume a 6¼ c and b 6¼ c. Then w(a, c; b, c)¼1, which impliesD(a, b; c, c), giving D4. &

An example of the situation of Lemma 1.3 is provided by remarks inAdeleke and Neumann (1998) and [19]. Let K be a field with a valuation vand value group L, and let E be the projective line over K, so E can beidentified with K[f1g. We define v(1)¼1. In an example of [19,

Sec. 7], it is noted inter alia that defining wða; b; c; dÞ ¼ v ðc�aÞðd�bÞðd�aÞðc�bÞ

� �gives

a projective valuation on E. On the other hand, an example of a D-relation on E is given by Adeleke and Neumann (1998, Theorem 30.4),after which it is noted that one way of defining D is: D(a, b; c, d),v(a� b)þ v(c� d) > v(a� d)þ v(b� c). It follows that D is obtained fromw as in 1.3. This example, of course, explains the name ‘‘projective valua-tion’’.

The main result, Theorem 3.8, is proved by first establishing theresult when E is finite, using another notion of generalised tree, whichwe call an order tree.

Definition 1.4. A partially ordered set (P, �), such that the set of prede-cessors of any element is linearly ordered, is called an order tree. (This is theterminology of Chiswell, 1998. An order tree is called a pseudotree in [16].Also, an order tree is called a lower semilinear ordering in Adeleke andNeumann, 1998, but we are assuming only their axioms A1, A2 and A3�.)

For example, if X is a L-tree, then we can define an order tree (X, �)by taking a basepoint x, and letting y� z if and only if y2 [x, z]. (SeeChiswell, 1994, beginning of Sec. 4.)

Theorem 3.8 is then proved in general using ultraproducts. This is atechnique used by the first author in Chiswell, (1994, 1998). Not allnotions of generalised tree have succumbed to this approach. We donot know of a natural way of viewing a profinite tree as a L-tree, andmedian algebras (see Bandelt and Hedlikova, 1983) seem too general tobe considered genuinely ‘‘treelike’’. (They can, however, be used in thestudy of generalised trees – see Basarab, 1997, 1998.)

Most of this paper is a revised version of part of the second author’sthesis ([15]).

2. PROJECTIVE VALUATIONS AND K-TREES

We begin with a precise definition of the projective valuation on theends of a L-tree and a proof that the axioms (PV1)–(PV3) are satisfied.

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To do this requires the idea of an end map. Given an end e of a L-tree X,an end map towards e is a map d :X!L such that, for all x2X, dj[x,e) isan isometry, with least value d(x). By Bass (1991, Lemma 1.4) such a mapexists and is unique up to an additive constant. We shall also make use ofthe linear tree (a, b) defined by two ends a, b and the point Y(a, b, c) asso-ciated to the distinct ends a, b, c, which is the unique point in the inter-section (a, b)\ (b, c)\ (c, a) (see Bass, 1991, Lemma 1.5). Note that, forany ends a, b, c, (a, b)\ (a, c)¼ [Y(a, b, c), a). For if x2 (a, b)\ (a, c), thenx2 (a, b)¼ [Y(a, b, c), a)[ [Y(a, b, c), b) (since Y(a, b, c)2 (a, b)).If x2 [Y(a, b, c), b), then X2 (b, c), so x2 (a, b)\ (a, c)\ (b, c), i.e.,x¼Y(a, b, c). Conversely, [Y(a, b, c), a)� (a, b)\ (a, c) since Y(a, b, c)2(a, b) and Y(a, b, c)2 (a, c).

Now let d be an end map towards b. We define, for distinct ends a, b,c, d, w(a, b; c, d)¼ d(Y((a, b, d))� d(Y(a, b, c)). Note that this is indepen-dent of the choice of d by Bass (1991, Lemma 1.4).

Lemma 2.1. In these circumstances:

(1) If w(a, b; c, d) > 0 then Y(a, b, c)¼Y(a, c, d).(2) w satisfies Axiom PV2.(3) If a, b, c and d are pairwise distinct ends of X, then (a, c)\

(b, d)¼; if and only if w(a, b; c, d) > 0.

Proof. (1) Let d be an end map towards b. If w(a, b; c, d) > 0 thend(Y(a, b, d)) > d(Y(a, b, c)) and it follows that [Y(a, b, c), a)~[Y(a, b, d), a) (see the proof of 1.4 and the observation at the start ofthe proof of 1.5 in Bass, 1991). In particular, Y(a, b, c)2 [Y(a, b, d), a)�(a, d). Hence Y(a, b, c)2 (a, d)\ (a, c)¼ [Y(a, c, d), a). Also, Y(a, b, c) 6¼Y(a, b, d). It also follows, since (a, d) is a linear tree, that Y(a, b, d)2[Y(a, b, c), d).

Next, suppose x2 [Y(a, b, c), c)\ [Y(a, b, c), d). Then

½x;Y ða; b; cÞ� \ ½Y ða; b; cÞ;Yða; b; dÞ� � ½Yða; b; cÞ; cÞ \ ða; bÞ¼ ða; cÞ \ ðb; cÞ \ ða; bÞ¼ fY ða; b; cÞg;

hence Y(a, b, c)2 [x, Y(a, b, d)]� [Y(a, b, c), d) (by the axioms for a L-tree), so x¼Y(a, b, c) since Y(a, b, c) is an endpoint of [Y(a, b, c), d). Thus[Y(a, b, c), c)\ [Y(a, b, c), d)¼fY(a, b, c)g, i.e., Y(a, b, c)2 (c, d). There-fore Y(a, b, c)2 (c, d)\ (c, a)¼ [Y(a, c, d), c).

We have shown that Y(a, b, c)2 [Y(a, c, d), a)\ [Y(a, c, d), c) and thisintersection is fY(a, c, d)g (sinceY(a, c, d)2 (a, c)), soY(a, b, c)¼Y(a, c, d).

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(2) Let d0 be an end map towards a. Suppose w(a, b; c, d) > 0.Then dþ d0 is constant on (a, b) (Bass, 1991, Lemma 1.5), sow(a, b; c, d)¼�d0(Y(a, b, d))þ d0(Y(a, b, c)) and similarly w(a, d; c, b)¼�d0(Y(a, d, b))þ d0(Y(a, d, c)), and these are equal by (1). Also,w(a, c; b, d)¼�d0(Y(a, c, d))þ d0(Y(a, c, b))¼ 0, again by (1).

(3) Suppose (a, c)\ (b, d)¼;; then Y(a, b, c) 6¼Y(a, b, d) sinceY(a, b, c)2 (a, c) and Y(a, b, d)2 (b, d), so w(a, b; c, d) 6¼ 0. Ifw(a, b; c, d) < 0 then (as in part (1)) [Y(a, b, c), b)� [Y(a, b, d), b), soY(a, b, c)2 [Y(a, b, d), b)� (b, d), hence Y(a, b, c)2 (a, c)\ (b, d), acontradiction. Hence w(a, b; c, d) > 0.

Conversely suppose w(a, b; c, d) > 0. Then (again as in part (1))

½Yða; b; dÞ; bÞ~ ½Y ða; b; cÞ; bÞ � ða; bÞ \ ðb; cÞ:

Hence Y(a, b, d) 6¼Y(a, b, c), and since (a, b)\ (b, c)\ (a, c)¼fY(a, b, c)g,it follows that [Y(a, b, d), b)\ (a, c)¼;.

By (2), w(a, d; b, c) > 0, so similarly [Y(a, b, d), d)\ (a, c)¼;.Since (b, d)¼ [Y(a, b, d), d)[ [Y(a, b, d), b), it follows that (a, c)\(b, d)¼;. &

Lemma 2.2. The mapping w defined before Lemma 2.1 is a projectivevaluation on the set E of ends of the L-tree X.

Proof. If d0 is an end map towards a, then dþ d0 is constant on (a, b)(Bass, 1991, Lemma 1.5), so w(a, b; c, d)¼�d0((Ya, b, d))þ d0(Y(a, b, c))and it follows that w(a, b; c, d)¼�w(b, a; c, d), and clearly w(a, b; c, d)¼�w(a, b; d, c).

Next, If w(a, b; c, d) > 0 then w(a, b; c, d)¼w(a, d; c, b) by Lemma2.1(2), and w(a, d; c, b)¼w(d, a; b, c)¼w(d, c; b, a) (previous remarks andLemma 2.1(2)). Then w(d, c; b, a)¼w(c, d; a, b) by the previous remarks,hence w(a, b; c, d)¼w(c, d; a, b).

If w(a, b; c, d) < 0 then w(a, b; d, c) > 0, so w(a, b; d, c)¼w(d, c; a, b)by the previous case, hence by the remarks at the beginning, w(a, b; c, d)¼w(c, d; a, b).

If w(a, b; c, d)¼ 0, then by Lemma 2.1(3) (a, c)\(b, d) 6¼ ;, that is(c, a)\ (d, b) 6¼ ;, so by Lemma 2.1(3) w(c, d; a, b)� 0. If w(c, d; a, b) <0, applying the previous case gives w(a, b; c, d)¼w(c, d; a, b) < 0,a contradiction, so w(c, d; a, b)¼ 0¼w(a, b; c, d). This shows PV1 issatisfied, and PV2 is by Lemma 2.1.

For PV3 consider the equivalent statement that if a, b, c, d, e2E arepairwise distinct, then w(d, e; a, b)þw(d, e; b, c)¼w(d, e; a, c). This clearlyfollows, on computing both sides via an end map towards e. &

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Our next aim is to prove a converse to Lemma 2.2, that given a pro-jective valuation w on a set E, E can be embedded in the set of ends ofsome L-tree so that w is the restriction to quadruples from E of the pro-jective valuation associated to the tree by Lemma 2.2. We follow theconstruction given in [19], but filling in the details is lengthy.

Let E be a set and w a projective valuation on E. For a, b2E anda 6¼ b let

Aða; bÞ ¼ fx : E n fa; bg ! LjxðdÞ � xðcÞ ¼ wða; b; c; dÞfor all d; c 2 E n fa; bgg

We assume E has at least three elements, so that this set is not empty.In fact, given a2L and e2Enfa, bg, there is a unique element x of A(a, b)satisfying x(e)¼ a, defined by x(c)¼ a�w(a, b; e, c) for c2Enfa, bg.It follows easily from PV1 and PV3 that x2A(a, b).

Note. Two functions in A(a, b) differ by a constant, and A(a, b) is iso-morphic as a set toL. (To get an isomorphism just fix e andmap x to x (e).)

The idea is that A(a, b) should represent (a, b), but we have to gluethese sets together appropriately to get a L-tree. To do this, define � tobe the equivalence relation on O¼S

A(a, b) (union over all pairs (a, b)with a 6¼ b) generated by two relations ’1 and ’2, defined as follows.Let x’1 y if and only if x2A(a, b), y2A(b, a) and x¼�y. Let x’2 yif and only if there are elements a, b, c of E with c 6¼ a, x2A(b, a),y2A(b, c) and x(c)¼ y(a)� 0. Before proceeding to the main construc-tion, we have to prove some technical lemmas on properties of �.

Lemma 2.3. Suppose x� y, so there is a sequence x¼ x0, x1, . . . , xn¼ ywhere either xi’1 xiþ1 or xi’2 xiþ1 for 0� i < n. If ’1 and ’2 do not alter-nate in this sequence, it can be replaced by a shorter sequence. Consequentlythere is such a sequence, no longer than the original, with ’1 and ’2 alter-nating.

Proof. We consider the ways in which x0, x1, . . . , xn can fail to alternate,showing how to shorten it in each case. If xi�1’1xi’1, xiþ1, thenxi�1¼�xi and xi¼�xiþ1, so xi�1¼ xiþ1 and this part of the sequencecan be replaced by just xi�1.

If xi�1’2 xi’2 xiþ1, then there are two possibilities. If both xi�1 andxiþ1 are elements of A(b, a) while xi2A(b, c), then xi�1(c)¼ xi(a)¼xiþ1(c), hence xi�1¼ xiþ1 and so the subsequence xi�1’2 xi ’2 xiþ1 canbe replaced by just xi�1.

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On the other hand if xi�12A(b, a), xi2A(b, c), and xiþ12A(b, e),with a, c, e all distinct, then all we know is that xi�1(c)¼ xi(a)� 0 andxi(e)¼ xiþ1(c)� 0. We claim that xi�1(e)¼ xiþ1(a)� 0, so the subsequencexi�1’2 xi’2 xiþ1 can be replaced by xi�1’2 xiþ1.

Let m¼ xi(a)� xi(e), which is also equal to xi�1(c)� xiþ1(c).By definition m¼w(b, c; e, a)¼w(e, a; b, c). There are now three

possibilities.

Case 1. m > 0. This means that by PV2we havew(b, c; e, a)¼w(b, a; e, c)and w(b, e; c, a)¼ 0. That ism¼ xi�1(c)� xi�1(e) and also xiþ1(a)¼ xiþ1(c).That is xi�1(e)¼ xi�1(c)�m¼ xi(e)¼ xiþ1(c)¼ xiþ1(a) and xi(e)� 0.

Case 2. m < 0. By PV1 we get w(b, c; a, e) > 0 and so again by PV2 weget �m¼w(b, c; a, e)¼w(b, e; a, c)¼ xiþ1(c)� xiþ1(a) and w(b, a; c, e)¼ 0i.e., xi�1(e)¼ xi�1(c). Therefore xiþ1(a)¼ xiþ1(c)þm¼ xi(e)þm¼ xi(a)¼xi�1(c)¼ xi�1(e) and xi(a)� 0.

Case 3. m¼ 0. So xi(a)¼ xi(e)¼ xi�1(c)¼ xiþ1(c). Also let

X¼w(b, a; c, e)¼ xi�1(e)� xi�1(c) and letY¼w(b, e; a, c)¼ xiþ1(c)� xiþ1(a)

If X < 0, then w(b, a; e, c)¼w(b, c; e, a) > 0 and w(b, e; a, c)¼ 0 byPV1 and PV2. But w(b, c; e, a)¼m¼ 0, so X� 0.

If X > 0 we have k¼w(b, a; c, e)¼w(b, e; c, a) > 0 i.e., �k¼Y < 0,so 0� xiþ1(c) < xiþ1(a) and since xi�1(c)¼ xiþ1(c), xi�1(e)� xiþ1(a)¼w(b, a; c, e)þw(b, e; a, c)¼ 0, by PV1, as we wanted.

If X¼ 0, we have xiþ1(c)¼ xi�1(c)¼ xi�1(e)¼ xi(e)¼ xi(a). That is,both w(b, a; c, e) and w(b, c; a, e) are equal to zero. Now assume thatxiþ1(c) 6¼ xiþ1(a), that is w(b, e; a, c) 6¼ 0. If w(b, e; a, c) > 0, then byPV2 we have w(b, c; a, e) > 0, giving us a contradiction. Also if w(b,e; a, c) < 0, then w(b, e; c, a) > 0, and w(b, a; c, e) > 0, by PV1 andPV2, which is a contradiction. Therefore xiþ1(c)¼ xiþ1(a)¼ xi�1(e)� 0.

This completes the proof. &

Lemma 2.4. If x’2 y’1�y’2 z’1�z then either x’1�x’2�z, orthere exists u2O such that x’2 u’1�u’2�z, or there exists u2O suchthat x’1�x’2 u’1�u’2�z.

Proof. Suppose x2A(a, b), y2A(a, d) and z2A(d, c), so b 6¼ d, a 6¼ c.Then

xðdÞ ¼ yðbÞ � 0 ð1Þ�yðcÞ ¼ zðaÞ � 0 ð2Þ

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Also, a, b, c, d are pairwise distinct except for the possibility b¼ c. Butthen by (1) and (2), �x(d)¼ z(a)¼ 0, so �x(d)¼�z(a)� 0, hencex’1�x’2�z.

Assume b 6¼ c. Then

xðdÞ � xðcÞ ¼ wða; b; c; dÞ ð3ÞyðbÞ � yðcÞ ¼ wða; d; c; bÞ ð4ÞzðaÞ � zðbÞ ¼ wðd; c; b; aÞ ð5Þ

By (1) and (2), w(a, d; c, b)¼ y(b)� y(c)� 0, giving two cases.

Case 1. w(a, d; c, b)¼ 0, i.e., y(b)¼ y(c). By (1) and (2), y(b)¼ y(c)¼ 0 so

xðdÞ ¼ zðaÞ ¼ 0 ð6ÞAlso, w(a, b; c, d)� 0, for if w(a, b; c, d) > 0 then w(a, d; c, b) > 0 by PV2,contrary to assumption. By (1) and (3), this implies x(c)� 0.

Define u2A(a, c) by u(b)¼ x(c), so x’2 u’1�u, and to prove�u’2�z, we have to show �u(d)¼�z(a)� 0, and by (2) we need to showthat u(d)¼ 0.

Now

uðbÞ � uðdÞ ¼ wða; c; d; bÞ ð7Þ

If w(c, a; d, b) > 0 then w(c, b; d, a) > 0 by PV2, but w(c, b; d, a)¼w(b, c; a, d)¼w(a, d; b, c)¼ 0 by assumption and PV1, a contradiction.

If w(c, a; d, b) < 0 then w(a, c; d, b) > 0 by PV1, so w(a, c; d, b)¼w(a, b; d, c)¼�w(a, b; c, d) by PV2 and PV1. By (3) and (7),u(b)� u(d)¼ x(c)� x(d), so u(d)¼ x(d)¼ 0 by (6).

If w(c, a; d, b)¼ 0 then u(b)¼ u(d) by (7). As already noted,w(a, b; c, d)� 0, and if w(a, b; c, d) < 0, then w(c, d; b, a) > 0 by PV1,so w(c, d; b, a)¼w(c, a; b, d) > 0 by PV2, hence w(c, a; d, b) < 0 byPV1, a contradiction. Hence w(a, b; c, d)¼ 0 and x(c)¼ x(d)¼ 0 by (3)and (6). Thus u(d)¼ u(b)¼ x(c)¼ 0.

Case 2. w(a, d; c, b) > 0. Then w(a, d; c, b)¼w(a, b; c, d) > 0 by PV2,so by (3) and (4), y(b)� y(c)¼ x(d)� x(c) hence by (2)

yðcÞ ¼ xðcÞ � 0 ð8Þ

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Also, w(a, b; c, d)¼�w(d, c; a, b)¼w(d, c; b, a) by PV1, so by (3) and (5),

zðaÞ � zðbÞ ¼ xðdÞ � xðcÞ¼ yðbÞ � yðcÞ by ð1Þ and ð8Þ

hence by (1) and (2)

yðbÞ ¼ �zðbÞ � 0: ð9Þ

Define u2A(b, c) by u(a)¼�x(c). By (8), u’2�x. We need to show�u’2�z, i.e., �u(d)¼�z(b)� 0. By (9), we need only show u(d)¼ z(b).Since u2A(b, c), u(d)� u(a)¼w(b, c; a, d)¼�w(a, d;c, b)¼ y(c)� y(b),using PV1 and (4). It follows by (8), (9) and the definition of u thatu(d)¼ z(b). &

Lemma 2.5. If x, y2E and x� y then there is a sequencex¼ x0’ � � � ’ xn¼ y with ’ either ’1 or ’2 and n� 4.

Proof. There is a sequence satisfying the first condition and we showthat, if n� 5, it can be replaced by a shorter sequence. For by Lemma2.3, we can replace it by an alternating sequence without increasing thelength. If this still has length greater than 5, then by Lemma 2.4, wecan either replace the sequence by a shorter one, or by one with two suc-cessive occurrences of ’1 or of ’2. Then by Lemma 2.3, we can againshorten the sequence. The lemma follows. &

The quotient O=� can now be made into a L-tree, and we first showhow to define the metric on it.

For c2Enfa, bg, the mapping A(a, b)!L, x 7! x(c) is a one to onecorrespondence. Also, if d2Enfa, bg, then x(c)� y(c)¼ x(d)� y(d), sodefining da, b(x, y)¼ jx(c)� y(c)j gives a metric on A(a, b) making it iso-morphic to L and which is independent of c.

Under the quotient map O!O=�, A(a, b) is mapped injectively to asubset which will be denoted by Aa,b. (It follows easily from Lemma 2.5that if x, y2A(a, b) and x� y then x¼ y.) Thus there is a metric on Aa,b

given by d([x], [y])¼ da,b(x, y)¼ jx(c)� y(c)j, where [x] denotes theequivalence class of x, and c is any element of Enfa, bg. This makesAa, b metrically isomorphic to L.

The metric on O=� will be defined so that its restriction to each Aa,b

is the metric just defined. We begin with two observations.

Remark 1. Given a2E and x2O, there exists x0 � x and b2E, b 6¼ asuch that x0 2A(a, b). To show this take x2A(c, d) for some c 6¼ d.

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If c¼ a then take x0 ¼ x and let b¼ d. If d¼ a, take x0 ¼�x2A(a, c) andb¼ d. Otherwise, if x(a)� 0, then x’2 y2A(c, a) where y(d)¼ x(a)and x0 ¼�y2A(a, c) so take b¼ c. If x(a) < 0, then x��x2A(d, c)and �x(a) > 0 so proceed as previously.

Remark 2. Any two points of O=� are in Aa, b for some a, b2E. Thatis, given x, y2O, there exists a, b2E, with a 6¼ b such that x� x0 andy� y0 for some x0, y0 2A(a, b). By Remark 1 we can find a, b, c2E, witha 6¼ b and a 6¼ c such that x2A(a, b) and y2A(a, c). If b¼ c we are fin-ished. If x(c)� 0, we have x’2x

0, where x0 2A(a, c) is the element withx0(b)¼ x(c) and we take y¼ y0. Similarly if y(b)� 0. Otherwisex’1�x’2 x

0, where x0 2A(b, c) satisfies x0(a)¼�x(c). Also y’1�y’2

z’1�z where z2A(c, b) satisfies z(a)¼�y(b) and we take y0 ¼�z2A(b, c), so y0(a)¼ y(b).

Now given u, v2O=�, by Remark 2 take x, y2A(a, b) for some a,b2E, a 6¼ b, such that u¼ [x] and v¼ [y] and define d(u, v)¼ da,b(x, y). Toshow d is well defined, first take x0, y0 2A(a, b0) such that [x]¼ [x0]¼ uand [y]¼ [y0]¼ v, where b0 6¼ b. It follows from Lemma 2.5, bearing inmind that the sequence of length at most 4 can be taken to be alternatingby the proof of Lemma 2.3, that x’2 x

0 and y’2 y0. It follows that

da,b(x, y)¼ jx(b0)� y(b0)j and da,b0(x0, y0)¼ jx0(b)� y0(b)j are equal.

Now take x0, y0 2A(a0, b) such that [x]¼ [x0]¼ u and [y]¼ [y0]¼ v.It follows from Lemma 2.5 that x’1�x’2�x0 ’1 x

0 and similarly fory, y0, hence da,b(x, y)¼ db,a(�x, � y)¼ db,a0(�x0, � y0)¼ da0,b(x

0, y0)This shows that d is well-defined. The proof that (O=�, d) is a L-tree

is part of the proof of the next theorem.

Theorem 2.6. Given a projective valuation w on a set E, we can find a L-tree X such that E can be embedded in the set of ends X1 of X, and w is theprojective valuation arising from X.

Proof. If jEj ¼ 2, we can clearly take X¼L, and if jEj < 2 we can take Xto be a single point. Otherwise, we take X¼O=� with d the function con-structed above. We first show that (X, d) is a L-tree.

Take three distinct points a, b, c2E. We claim that Aa,b\Aa,c\Ab,c

is a single point, which we denote by Y(a, b, c). For suppose x� y� z,where x2A(a, b), y2A(a, c) and z2A(b, c). By Lemma 2.5, x’2 y, sox(c)¼ y(b)� 0. Also, �y�� z, so by Lemma 2.5, �y’2�z, that is,�y(b)¼�z(a)� 0. Hence x(c)¼ y(b)¼ z(a)¼ 0, showing there is at mostone point in the intersection. On the other hand, if x is the elementof A(a, b) with x(c)¼ 0, y2A(a, c) is the element with y(b)¼ 0 and

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z2A(b, c) with z(a)¼ 0, then x’2 y and y’1�y’2�z’1 z, so[x]¼ [y]¼ [z] is in the intersection.

Now let Y(a, b, c)¼ [w]¼ [w0], where w2A(a, b) and w0 2A(a, c). Weclaim that d([x], [y])¼ da,b(x, w)þ da,c(y, w0) for x2A(a, b) and[y]2Aa,cnAa,b, where y2A(a, c). Note that y(b) < 0, otherwise[y]¼ [y0]2Aa,b, where y0 is the element of A(a, b) with y0(c)¼ y(b), con-trary to assumption.

If x(c) < 0, then as in Remark 2, x� x0, where x0 2A(b, c) is the ele-ment with x0(a)¼�x(c), and y� y0, where y0 2A(b, c) satisfies y0(a)¼ y(b).Thus d([x], [y])¼ db,c(x

0, y0)¼ jx0(a)� y0(a)j ¼ j� x(c)� y(b)j ¼�x(c)�y(b)¼ da,b(x, w)þ da,c(y, w

0). If x(c)� 0, then x� x0, where x0 2A(a, c)satisfies x0(b)¼ x(c), so d([x], [y])¼ da,c(x

0, y)¼ jx0(b)� y(b)j ¼ x0(b)�y(b)¼ x(c)� y(b)¼ da,b(x, w)þ da,c(y, w

0).It follows that if u2Aa,b and p 62Aa,b say p2Aa,c (using Remark 1

above), that d(p, u)¼ d(p, w)þ d(w, u), where w¼Y(a, b, c).We now show that d is a metric. Clearly d satisfies all the axioms for a

metric, except perhaps the triangle inequality. So take u, v, p2O=� .By Remark 2 we can assume u, v2Aa,b for some a, b2E. If p2Aa,b weare done, because d restricted to Aa,b is a metric. Otherwise d(p, u)¼d(p, w)þ d(w, u) and d(p, v)¼ d(p, w)þ d(w, v), where w¼Y(a, b, c)and p2Aa,c. So adding d(p, u)þ d(p, v)¼ 2d(p, w)þ d(u, w)þ d(w, v)�d(w, u)þ d(w, v)� d(u, v) again because d restricted to Aa,b is a metric.

Because Aa,b is metrically isomorphic to L, this is a geodesic metricspace. To be specific for x, y2A(a, b) there is a segment [u, v]Aa,b, where[x]¼ u and [y]¼ v. (Also [z]2 [u, v], z(c)2 [u(c), v(c)]Lwhere c2Enfa, bg.)

We claim that this geodesic is unique. So assume there is another geo-desic with endpoints u, v containing a point p 62Aa,b. Hence from aboved(p, u)¼ d(p, w)þ d(w, u) and similarly d(p, v)¼ d(p, w)þ d(w, v), andadding we obtain d(u, v)¼ 2d(p, w)þ d(u, w)þ d(w, v)� 2d(p, w)þd(u, v) which implies that d(p, w)¼ 0, i.e., p2Aa,b, a contradiction.

Next, we show that for any three points u, v, w the intersection[u, v]\ [v, w]\ [w, u] is a point which we shall denote by Y(u, v, w). If thereis a set Aa,b such that all three of u, v, w2Aa,b then we are done, as Aa,b isa L-tree, being isomorphic to L. By the remarks above, we can assume u,v2Aa,b and w2Aa,c, where a, b, c are distinct elements of E. It followsfrom Remark 2 above that u2Aa,c or u2Ab,c and similarly for v.Now w¼ [z] for z2A(a, c) and z(b) < 0, since w 62Aa,b. Thenz’1�z’2 z

0 ’1�z0, where z0 2A(c, b) satisfies z0(a)¼�z(b), so �z0 2A(b, c), hence w2Ab,c as well. If both of u, v2Aa,c, then u, v, w2Aa,c,contrary to our assumption. Similarly they can’t both be in Ab,c. So with-out loss of generality assume u2Aa,cnAb,c and v2Ab,cnAa,c. Now [u, v]\[v, w]\ [w, u]�Aa,b\Aa,c\Ab,c¼fY(a, b, c)g.

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Now u¼ [x], where x2A(a, c) and x(b)� 0, since u2Aa, b. Alsow¼ [z], where z2A(a, c) and z(b) < 0. By its definition Y(a, b, c)¼ [y],where y2A (a, c) and y(b)¼ 0. It follows that Y (a, b, c)2 [u, w]. Similararguments show that Y(a, b, c)2 [v, w] and Y(a, b, c)2 [u, v].

We want to show that [u, v]\ [u, w]¼ [u, Y(a, b, c)]. If u, v, w2Aa,b

then we are done, as Aa, b is a L-tree. So as before we can assume thatu2Aa,bnAb,c, v2Aa,bnAa,c and w2Aa,cnAa,b. This implies that u2Aa,c.Now Y(a, b, c)¼ [x], where x2Aa,b and x(c)¼ 0. Also Y(a, b, c)¼ [y],where y2Aa,c and y(b)¼ 0. So if s2 [u, v]\ [u, w], then s¼ [x0] withx0 2A(a, b) and s¼ [y0] with y0 2A(a, c) such that x0 ’2 y

0 andx0(c)¼ y0(b)� 0. This applies to s¼ u as well, and so s2 [u, Y(a, b, c)],from the description of this segment given above.

If [u, w]\ [w, v]¼fwg we have w¼Y(u, v, w) and so w2 [u, v]Aa, b,hence [u, v]¼ [u, w][ [w, v].

We have now shown that (X, d) is a L-tree.Let a2E and let u2X. Then by Remark 1, u2Ab,a¼Aa,b for some

b2E, b 6¼ a, say u¼ [x], where x2A(b, a). If c 6¼ b, a, then Ab,a!L,u 7! x(c) is a metric isomorphism, and the inverse image of [x(c), 1)Lis then an X-ray with endpoint u, say R. Since two elements of A(b, c) dif-fer by a constant, the end determined by R does not depend on the choiceof u or of c. Also, if u0 2Ab0, a similarly defines the X-ray R0, then R and R0

define the same end, since they both define the same end as the X-raydetermined by Y(a, b, b0). So this associates an end a of X with a, anddefines a mapping E!X1, a 7! a.

If a, b2E, a 6¼ b, then a 6¼ b�. In fact Aa, b¼ (a, b�). So we have anembedding E!X1, and we identify a and a. This justifies the use ofthe notation Y(a, b, c) earlier in the proof. Also, the mapping u 7! x(c)above is an end map towards a.

Let w0 be the projective valuation defined by the L-tree (X, d)(Lemma 2.2) and take four distinct points a, b, c, d2E. Let Y(a, b, c)¼[x] where x2A(a, b) and let Y(a, b, d)¼ [y] where y2A(a, b), sox(c)¼ y(d)¼ 0. Then w0(a, b;c, d)¼ x(c)� y(c)¼�y(c)¼ y(d)� y(c)¼w(a, b;c, d) by the definition of A(a, b). This completes the proof. &

In Theorem 2.6 it is not always possible to find a L-tree X such thatE¼X1. For a counterexample, see [15].

3. D-RELATIONS

In this section we shall construct a L-tree from an abstract directionset (E, D), by first constructing a projective valuation defining it, then

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using the results of Sec.2. We begin by stating an elementary consequenceof the axioms.

Lemma 3.1. Let (E, D) be an abstract direction set. Then

(D9) D(a, b; c, d)^D(a, b; d, e))D(a, b; c, e)

Proof. This is proved just before Theorem 22.1 in Adeleke andNeumann (1998). &

Given an abstract direction set (E, D), we shall construct an ordertree from it. This is essentially done in Adeleke and Neumann (1998,Theorem 12.4 and Theorem 22.1). However, it involves the construc-tion of a ‘‘C-relation’’ from D, which omits a chosen basepoint in E.Also, to obtain an order tree, we would need to replace the partialordering in Adeleke and Neumann (1998, Theorem 12.4) by its dual.(Our notion of order tree is chosen for consistency with Chiswell(1994), [14], [17].) We shall therefore give direct proofs of the nexttwo lemmas.

Let (E, D) be an abstract direction set. Choose a basepoint x2E.Define a binary relation on the set E2 of unordered pairs of elementsof E by the following: For all a, b, c, d2E,

fc; dg fa; bg if and only if :Dða; x; c; dÞ ^ :Dðb; x; c; dÞ:

Lemma 3.2. The relation is a pre-order on E2.

Proof. To show it is reflexive, we need to show :D(a, x; a, b)^:D(b, x;a, b) and by D1 we need only show :D(a, x; a, b). If x 6¼ a, a 6¼ b thenD(b, x; a, a) by D4, and by D1, :D(b, a; x, a), and further use of D1 gives:D(a, x; a, b). Otherwise, as noted after Definition 3.1, :D(a, x; a, b).

To show transitivity we adapt the proof of Adeleke and Neumann(1998, Lemma 12.1). Assume that fe, fgfc, dgfa, bg. This meansthat

:Dða; x; c; dÞ ^ :Dðb; x; c; dÞ ^ :Dðc; x; e; f Þ ^ :Dðd; x; e; f Þholds. We need to show that :D(a, x; e, f )^:D(b, x; e, f ) holds as well.Suppose (for a contradiction) that D(a, x; e, f) holds. By D1 and D3,

Dða; x; e; f Þ ^ :Dðc; x; e; f Þ ) Dða; x; c; f ÞDða; x; e; f Þ ^ :Dðd; x; e; f Þ ) Dða; x; d; f Þ

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and by D1 and D9 (Lemma 3.1)

Dða; x; c; f Þ ^Dða; x; d; f Þ ) Dða; x; c; dÞ:Since :D(a, x; c, d) holds by assumption, :D(a, x; e, f ) holds. Similarly,interchanging a and b, :D(b, x; e, f ) holds. &

Using standard facts about pre-orders, we obtain an equivalencerelation � on E2 by defining fa, bg�fc, dg if and only if fa, bgfc, dgand fc, dgfa, bg. Let P be the quotient E2=�. Let ha, bi denotethe equivalence class of fa, bg for fa, bg2E2. Also, we can define a par-tial ordering � on P by ha, bi� hc, di iff fa, bgfc, dg.

Lemma 3.3. For an abstract direction set (E, D) and basepoint x, let P beE2=�. With the partial ordering � above, the pair (P, �) forms an ordertree, with least element hx, xi.

Proof. As observed after Definition 3.1, :D(c, x; x, x) holds for allc2E, and it follows that hx, xi� ha, bi for all a, b2E.

We have to show that

ðhc; di � ha; biÞ ^ ðhe; f i � ha; biÞ ) ðhc; di � he; f iÞ _ ðhe; f i � hc; diÞso the assumption is that

:Dða;x; c; dÞ ^ :Dðb;x; c; dÞ ^ :Dða;x; e; f Þ ^ :Dðb;x; e; f Þ:Assume the conclusion is false, i.e.,

ðDðc;x; e; f Þ _Dðd;x; e; f ÞÞ ^ ðDðe;x; c; dÞ _Dðf ;x; c; dÞÞ:Using distributivity of ^ over _, this gives a disjunction of four possibi-lities, and we show each contradicts the assumption.

The first case isD(c, x; e, f )^D(e, x; c, d ). Now by D3,D(c, x; e, f ))D(c, x; e, a)_D(a, x; e, f ) and since :D(a, x; e, f ) by assumption,D(c, x; e, a) holds. Also by D3, D(e, x; c, d ))D(e, x; c, a)_D(a, x; c, d )and since :D(a, x; c, d ) by assumption, D(e, x; c, a) holds. But by D1and D2, D(e, x; c, a))D(e, x; a, c)):D(e, a; x, c)):D(c, x; e, a).Thus D(c, x; e, a)^:D(c, x; e, a) holds, a contradiction.

The next case is obtained by interchanging e and f in this argument,then the others by interchanging c and d in the first two. This does notchange the assumption by D1. &

Note. If (P, �) is an order tree which is discrete (i.e., for all x, y2P withx� y, the interval fz2P; x� z� yg is finite), and with least element p,

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then P can be embedded as the set of vertices of a simplicial tree by thefollowing method. Join two elements x, y2P by an edge when x < y, butthere is no element z2P such that x < z < y, where < is the strict par-tial order corresponding to �. Then the ordering is that induced by thesimplicial tree (or its corresponding Z-tree) as in the example after Defini-tion 3.5, using p as basepoint. (see [17] and Sec. 2 in [14])

Before stating the next lemma, we recall (Alperin and Bass, 1987, Sec.2.23) that a closed end of a L-tree is one defined by an endpoint e, whoserays are the segments [x, e] for x2X.

Lemma 3.4. Let (E, D) be an abstract direction set, with basepoint x andP equal to E=� as above. Assume the order tree (P, �) is discrete. Then Pcan be embedded as the set of points of a Z-tree in such a way that the pairsof the form ha, ai form the set of closed ends of the tree. Further, the map-ping a 7! ha, ai embeds the elements of E as the set of closed ends of thetree.

Proof. From the note above, the elements ha, bi 2P form the set ofpoints of a Z-tree. (In fact these points will turn out to be ha, bi¼Y(ha, ai, hb, bi, hx, xi), where Y is as defined earlier for ends.) To showthat ha, ai is actually an endpoint consider the following. Assume a 6¼ xand fa, agfb, cg. Then :D(b, x; a, a)^:D(c, x; a, a). Then by D4we get b¼ a and c¼ a. So ha, ai is a maximal element for all a, apart froma¼ x, for which hx, xi� hb, ci for all b, c holds. So ha, ai is a closed end ofthe Z-tree (X, d) for all a2Enfxg.

Note that ha, bi� ha, ai for all a and b, that is, :D(a, x; a, b), whichfollows from D4, D2 and the observation after Definition 3.1.

Now assume hx, xi is not an endpoint; by the previous paragraph,this means that there are at least two paths going out from it, that leadto distinct ends ha, ai and hb, bi, and so the point ha, bi, which belongsto both paths, must equal hx, xi. Now assume ha, bi¼ hx, xi. Since hx, xiis the least element of (P, �), this means that D(x, x; a, b), whichimplies by D4 and D1 that either a¼ x or b¼ x, which is a contradic-tion, and so there cannot be two paths going out from hx, xi.

Assume ha, ai¼ hb, bi. This means that :D(b, x; a, a)^:D(a, x; b, b),which implies that a¼ b. Finally if ha, bi is not equal to hc, ci for some c,then hx, xi < ha, bi < ha, ai, so ha, bi is not an endpoint. This shows theE embeds as the set of closed ends of the tree. &

Let (P, �) be a discrete order tree with least element x. Then by thedefinition of the corresponding simplicial tree, [x, p]¼fa2Pjx� a� pg

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for p2P. Now for a, b2P, [x, a]\ [x, b]¼ [x, c] for some c and we denotec by m(a, b), so m(a, b)¼min fa, bg¼Y(x, a, b).

Lemma 3.5. In these circumstances, let D be the D-relation on thecorresponding simplicial tree (viewed as a Z-tree). Then D(a, x; b, c),m(a, c) < m(b, c).

Proof. By the remarks after Lemma 1.3 D(a, x; b, c), [a, x]\ [b, c]¼;.Since m(a, c), m(b, c)� c, they are comparable, and m(b, c)� m(a, c)� cimplies m(b, c)2 [a, x]\ [b, c]. Conversely, if [a, x]\ [b, c]¼; then bythe Y-proposition (Alperin and Bass, 1987, 2.11), [x, a]\ [x, c]¼[x, m(a, c)]~ [b, x]\ [c, x]¼ [x, m(b, c)]. &

Lemma 3.6. In the situation of Lemma 3.4, the D-relation D0 arising fromthe Z-tree (see the remarks after Lemma 1.3), equals the original D-relationD on E.

Proof. Let (E, D) be an abstract direction set with chosen basepoint xand let (P, �) be the corresponding order tree, which we assume is dis-crete. By Lemma 3.4, we can identify a2E with ha, ai 2P. Then in thenotation of Lemma 3.5, m(a, b)¼ha, bi, because

ðhc; di � ha; aiÞ ^ ðhc; di � hb; biÞ , ðhc; di � ha; biÞ

since both statements are equivalent to :D(a, x; c, d)^:D(b, x; c, d).

Thus by Lemma 3.5,

D0(a, x; b, c), m(a, c) < m(b, c)

, :(m(a, c)� m(b, c))

(m(a, c) and m(b, c) are comparable, as noted in proving Lemma 3.5)

, :(hb, ci� ha, ci), D(a, x; b, c)_D(c, x; b, c) by the definition of (P, �).

But D(c, x; b, c) is false, by D1, D4 and the remarks after Definition3.1, so D0(a, x; b, c),D(a, x; b, c).

It will now follow that D¼D0 provided one can show the followinglemma.

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Lemma 3.7. If D is a D-relation on E and x2E, then

Dða; b; c; dÞ , ðDða;x; c; dÞ ^Dðc; d; b; xÞÞ_ ðDða; b; c; xÞ ^Dða; b; d; xÞÞ

Proof. Using D1 and D3,

Dða;b;c;dÞ)Dðb;a;c;dÞ^Dða;b;d;cÞ) ðDða;x;c;dÞ_Dða;b;c;xÞÞ^ðDðc;d;b;xÞ_Dða;b;d;xÞÞ

) ðDða;x;c;dÞ^Dðc;d;b;xÞÞ ðiÞ_ðDða;x;c;dÞ^Dða;b;d;xÞÞ ðiiÞ_ðDða;b;c;xÞ^Dða;b;d;xÞÞ ðiiiÞ_ðDða;b;c;xÞ^Dðc;d;b;xÞÞ ðivÞ

The possibilities in (i) are illustrated by:

The right hand possibility is in fact (ii) and we shall show that (ii) )(i). Assume (ii). First, by D1 and D3,

Dða; x; c; dÞ ) Dða; x; c; bÞ _Dðc; d; b; xÞ

and it suffices to show that D(a, x; c, b) is false. By D1 and D3 we have

Dða; b; d; xÞ ) Dðb; a; x; dÞ ) Dðc; a; x; dÞ _Dðb; a; x; cÞ

Now if D(b, a; x, c) were true, then this would imply that D(a, b; x, c) istrue by D1, so D(a, x; b, c) is false by D2, hence by D1 D(a, x; c, b) isfalse. Thus it suffices to show D(c, a; x, d) is false. But this follows byD1 and D2, since D(a, x; c, d) is true.

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We can interchange a with c and b with d in this argument, to show,using D1, that (iv) ) (iii). This proves the implication ) of the lemma.

To show the converse, it is enough to show that D(a, x; c, d)^D(c, d;b, x) implies D(a, b; c, d), as the other half can be shown by the sameargument, interchanging a with c and b with d, as before. AssumeD(a, x; c, d)^D(c, d; b, x). By D1 and D3,

Dða; x; c; dÞ ) Dðx; a; c; dÞ ) Dða; x; c; bÞ _Dða; b; c; dÞ ð1ÞDðc; d; b; xÞ ) Dðd; c; b; xÞ ) Dðd; c; b; aÞ _Dða; c; b; xÞ ð2Þ

By D2 and (2), D(a, x; c, b)):D(a, c; x, b))D(d, c; b, a), which impliesD(a, b; c, d) by D1. Thus by (1) D(a, b; c, d) holds. &

Theorem 3.8. Let E be a set and D a D-relation on E. Then we can definea projective valuation w on E such that D(a, b; c, d) is true if and only ifw(a, c; b, d) > 0.

Proof. Let x2E, and as before define an order tree (P, �) with least ele-ment hx, xi, which we identify with x. Let S be the set of all finite subsetsof E that contain x. For each finite subset F�E, with x2F, by restrictingD we have by Lemma 3.4 an order tree (PF, �) and a corresponding Z-tree, such that F embeds in PF as the set of closed ends. Also PF embedsin P in an order preserving way, taking ha, biF 7! ha, biE. We identifyha, biF with ha, biE. By Lemma 3.6, there is a projective valuation wF

on F, taking values in Z[f±1g such that, for all a, b, c, d2F,

Dða; c; b; dÞ , wF ða; b; c; dÞ > 0:

For F2S let AF¼fF 0 2SjF�F 0g. If F, F 0 2S, then AF\AF 0 �AF[F 0. Sothere is an ultrafilter U on S containing AF, for all F2S. Let �Z denotethe ultrapower (

QF2SZ)=U and let E¼ (

QF2SF)=U. There is a map

w ¼�Y

F2SwF

�.U : �EE4 ! Z [ f 1g;

which is again a projective valuation, by the Theorem of Łos (see van denDries and Wilkie, 1984).

Define fF: E!F by

fF ðeÞ ¼ e if e 2 Fx otherwise

�

and define f: E! E by f(e)¼ (fF (e))F2S=U, (the image of the family(fF (e))F2S in the ultraproduct E ).

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Then f is one to one. For suppose f(y)¼f(z). Then fF(y)¼fF(z)for all F2A, for some A2U. Now Afx,y,zg 2U, so A\Afx,y,zg 2U henceA\Afx,y,zg 6¼ 0.ChooseF2A\Afx,y,zg. Theny¼fF(y)¼fF(z)¼ z, soy¼ z.

For a, b, c, d2E we want

Dða; c; b; dÞ , � wðfðaÞ;fðbÞ;fðcÞ;fðdÞÞ > 0:

We know that

�wðfðaÞ;fðbÞ;fðcÞ;fðdÞÞ> 0, wF ðfF ðaÞ;fF ðbÞ;fF ðcÞ;fF ðdÞÞ> 0

for almost all F.Now Afx,a,b,c,dg 2U, and for F2Afx,a,b,c,dg we have

wF ðfF ðaÞ;fF ðbÞ;fF ðcÞ;fF ðdÞÞ ¼ wF ða; b; c; dÞand wF(a, b; c, d) > 0,D(a, c; b, d), as already noted. So D(a, c; b, d) istrue implies wF(a, b; c, d) > 0, which implies that

wF ðfF ðaÞ;fF ðbÞ;fF ðcÞ;fF ðdÞÞ > 0

for all F2Afx,a,b,c,dg, which implies that �w(f(a), f(b); f(c), f(d)) > 0.Conversely if

�wðfðaÞ;fðbÞ;fðcÞ;fðdÞÞ > 0

then

wF ðfF ðaÞ;fF ðbÞ;fF ðcÞ;fF ðdÞÞ > 0

for all F2A for some A2U. Now A\Afx,a,b,c,dg 6¼ ; since it belongs to U.So choose F2A\Afx,a,b,c,dg. Then

wF ða; b; c; dÞ ¼ wF ðfF ðaÞ;fF ðbÞ;fF ðcÞ;fF ðdÞÞ > 0;

which implies thatD(a, c; b, d) is true. Thus we obtain the required projec-tive valuation w by defining w(a, b; c, d)¼ �w((f(a), f(b); f(c), f(d)).

&

Corollary 3.9. Let (E, D) be an abstract direction set. Then we can embedthe elements of E as ends in a L-tree (X, d) in such a way that the projectivevaluation w arising from (X, d) defines D.

Proof. This is immediate from Theorem 2.6 and Theorem 3.8. &

Suppose a group G acts on an abstract direction set (E, D) (i.e., Gacts on E and preserves D). In Corollary 3.9 it would be desirable to

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conclude that G acts as isometries on (X, d). However to ensure this, arestrictive hypothesis seems necessary.

Call a subset of E G-finite if it is a finite union of G-orbits. Assumethat, for all G-finite subsets F of E containing x, the order tree (PF, �)is discrete. Then G acts as isometries on the corresponding Z-tree, so pre-serves wF. Replacing S in the proof of Theorem 3.8 by the set of allG-finitesubsets of E containing x, with suitable modifications we obtain a projec-tive valuation w on E defining D as in Lemma 1.3 and preserved by G.

Then in the construction of Theorem 2.6, there is an action of G onthe set O, where if x2A(a, b) and g2G, gx(c)¼ x(g�1c) (gx2A(ga, gb)).This action preserves ’1 and ’2, so preserves � and there is an action ofG on X¼O=�. Also, da,b(x, y)¼ dga,gb(gx, gy), hence G acts as isometrieson the L-tree (X, d) in Theorem 2.6.

REFERENCES

Adeleke, S. A., Neumann, P. M. (1998). Relations Related To Between-ness: Their Structure and Automorphisms. Vol. 623. Memoirs Amer.Math. Soc., Providence, RI: Amer. Math. Soc.

Alperin, R. C., Bass, H. (1987). Length functions of group actions on L-trees. In: Gersten, S. M., Stallings, J. R., eds. Combinatorial GroupTheory and Topology. Vol. 111. Annals of Math. Studies, Princeton:Princeton University Press, pp. 265–378.

Bandelt, H.-J., Hedlikova, J. (1983). Median algebras. Discrete Math.45:1–30.

Basarab, S. A. (1997). Directions and foldings on generalized Trees.Fund. Inform. 30:125–149.

Basarab, S. A. (1998). On discrete hyperbolic arboreal groups. Comm.Algebra 26:2837–2865.

Bass, H. (1991). Group actions on non-archimedean trees. In: Alperin,R. C., ed. Arboreal Group Theory. Vol. 19. New York: MSRI Pub-lications, Springer-Verlag, pp. 69–131.

Bennett, C. D. (1990). Affine L-buildings. Ph.D. thesis, University ofChicago.

Bowditch, B. H. (1999). Treelike Structures Arising from Continua andConvergence Groups. Vol. 662. Memoirs Amer. Math. Soc., Provi-dence, RI: Amer. Math. Soc.

Chiswell, I. M. (1994). Generalised trees and L-trees. In: Duncan, A. J.,Gilbert, N. D., Howie, J., eds. Combinatorial and Geometric GroupTheory, Edinburg 1993. Vol. 204. London Math. Soc. LectureNotes, Cambridge: Cambridge University Press, pp. 43–55.

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Chiswell, I. M. (1998). Protrees and L-trees. In: Kropholler, P. H., Niblo,G. A., Stohr, R., eds. Geometry and Cohomology in Group Theory.Vol. 252. London Math. Soc. Lecture Notes, Cambridge: Cam-bridge University Press, pp. 74–87.

Dunwoody, M. J. (1997). Groups acting on protrees. J. London Math.Soc. 56:125–136.

Herrlich, F. (1991). Moduli for stable marked trees of projective lines.Math. Ann. 291:643–661.

Hoare, A. H. M. (1988). Pregroups and length functions. Math. Proc.Cambridge Philos. Soc. 104:21–30.

Koudels, E. (1997). Group Actions on L-Trees. Ph.D. thesis. Universityof London.

Nikiel, J. (1989). Topologies on Pseudo-trees and Applications. Vol. 416.Memoirs Amer. Math. Soc., Providence, RI: Amer. Math. Soc.

Rimlinger, F. S. (1987). Pregroups and Base-Serre Theory. Vol. 361.Memoirs Amer. Math. Soc., Providence, RI: Amer. Math. Soc.

Shalen, P. B. (1991). Dendrology and its applications. In: Ghys, E.,Haefliger, A., Verjovsky, A., eds. Group Theory from a GeometricalViewpoint. Singapore: World Scientific.

Tits, J. (1986). Immeubles de type affine. In: Buildings and the Geometryof Diagrams, Como 1984. Vol. 1181. Lecture Notes in Mathematics,Berlin, Heidelberg, New York: Springer-Verlag, pp. 159–190.

van den Dries, L., Wilkie, A. (1984). Gromov’s theorem on groups ofpolynomial growth and elementary logic. J. Algebra 89:349–374.

Received April 2002Revised July 2002

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