Construction of isolated left orderings via partially central cyclic amalgamation
aa r X i v : . [ m a t h . G R ] F e b CONSTRUCTION OF ISOLATED LEFT ORDERINGS VIAPARTIALLY CENTRAL CYCLIC AMALGAMATION
TETSUYA ITO
Abstract.
We give a new method to construct isolated left orderings ofgroups whose positive cones are finitely generated. Our construction usesan amalgamated free product of two groups having an isolated ordering. Weconstruct a lot of new examples of isolated orderings, and give an exampleof isolated left orderings having various properties which previously knownisolated orderings do not have. Introduction
A total ordering < G on a group G is a left ordering if g < G g ′ implies hg < G hg ′ for all g, g ′ , h ∈ G . The positive cone of a left ordering < G is a sub-semigroup P ( < G ) of G consisting of < G -positive elements.The set of all left orderings of G is denoted by LO( G ). For g ∈ G , let U g be asubset of LO( G ) defined by U g = { < G ∈ LO( G ) | < G g } . We equip a topology on LO( G ) so that { U g } g ∈ G is an open sub-basis of the topology.This topology is understood as follows. For a left ordering < G of G , G is decomposedas a disjoint union G = P ( < G ) ⊔ { } ⊔ P ( < G ) − using the positive cone P ( < G ).Conversely, a sub-semigroup P of G having this property is a positive cone of aleft ordering of G : An ordering < P defined by g < P g ′ if g − g ′ ∈ P is a left-ordering whose positive cone is P . Thus LO( G ) is identified with a subset ofthe powerset 2 G −{ } . The topology of LO( G ) defined as above coincides with therelative topology as the subspace of 2 G −{ } , equipped with the powerset topology.In this paper, we always consider countable groups, so we simply refer a countablegroup as a group unless otherwise specified. Then it is known that LO( G ) is acompact, metrizable, and totally disconnected [10]. Moreover, LO( G ) is eitheruncountable or finite [5]. Thus as a topological space, LO( G ) is rather similar tothe Cantor set: The main difference is that the space LO( G ) might be non-perfect,that is, LO( G ) might have isolated points. Indeed, if LO( G ) has no isolated pointsand is not a finite set, then LO( G ) is homeomorphic to the Cantor set. We call aleft ordering which is an isolated point of LO( G ) an isolated ordering .It is known that a left ordering < G whose positive cone is a finitely generatedsemigroup is isolated. In this paper we will concentrate our attention to studysuch an isolated ordering. We say a finite set of non-trivial elements of G , G = { g , . . . , g r } defines an isolated left ordering < G of G if the positive cone of < G isgenerated by G as a semigroup. For an isolated left ordering < G of a group G , the Mathematics Subject Classification.
Primary 20F60 , Secondary 06F15.
Key words and phrases.
Orderable groups, isolated ordering, space of left orderings. rank of < G is the minimal number of generating sets of the positive cone of < G and denoted by r ( < G ). (If P ( < G ) is not finitely generated semigroup we define r ( < G ) = ∞ ).We say an isolated ordering < G of G is genuine if LO( G ) is not a finite set.Then LO( G ) contains (uncountably many) non-isolated points. The classificationof groups having non-genuine isolated orderings, namely, the classification of groupshaving finitely many left-orderings is given by Tararin (see [4]). On the other hand,it is difficult to construct genuine isolated left orderings, and few examples areknown. At this time of writing, as long as author’s knowledge, there are essentiallyonly two families of genuine isolated left orderings.(1) Dubrovina-Dubrovin ordering [1],[2].Let σ , . . . , σ n − be the standard generator of the n -strand braid group B n . The Dubrovina-Dubrovin ordering < DD is an isolated left ordering of B n whose positive cone is generated by { a , . . . , a n − } , where a i is givenby a i = ( σ n − i σ n − i +1 · · · σ n − ) ( − i The rank of the Dubrovina-Dubrovin ordering < DD is n −
1. See [1], [2] fordetails.(2)
Isolated orderings of Z ∗ Z Z [3],[8].Let G = Z ∗ Z Z be the group obtained as an amalgamated free productof two infinite cyclic groups over Z . Thus, G is presented as G = h x, y | x m = y n i . by using some integers m and n . Then the generating set { xy − n , y } de-fines an isolated left ordering < A of G , which is genuine if ( m, n ) = (2 , < A is 2. This example was found by Navas [8] for the case m = 2, and by the author [3] for general cases. We here remark that if( m, n ) = (2 ,
3) then G m,n is the 3-braid group B , and the isolated order-ing < A is the same as the Dubrovina-Dubrovin ordering < DD .Thus, it is desirable to find more examples or general constructions of isolatedleft orderings.In author’s previous paper [3], we gave one general method to construct isolatedorderings by using rather combinatorial approach. Following [8], we introduceda notion of Dehornoy-like ordering . This is a left-ordering whose positive coneconsists of certain kind of words over a special generating set S of G , which wecalled σ ( S ) -positive words . A Dehornoy-like ordering is a generalization of the De-hornoy ordering of the braid groups, one of the most interesting left orderings: TheDehornoy ordering has various stimulating features and a lot of interesting inter-pretations that relate many aspects of braid groups and orderings. See [1] for thetheory of the Dehornoy ordering. One fascinating property of the Dehornoy order-ing is that one get the Dubrovina-Dubrovin ordering by modifying the Dehornoyordering. ONSTRUCTION OF ISOLATED LEFT ORDERINGS 3
We showed that, under some condition which we called the Property F , Dehornoy-like orderings and the Dehornoy ordering share various properties. In particular, wehave shown that a Dehornoy-like ordering produces an isolated ordering and viceversa. Indeed, it is shown that the above two families of known isolated orderingsare derived from Dehornoy-like orderings.However it seems to be more difficult to find an example of a Dehornoy-likeordering than to find an example of an isolated ordering directly, since the definitionof Dehornoy-like orderings includes complicated combinatorics.The aim of this paper is to give a new construction of isolated left orderingsby means of the partially central cyclic amalgamation . From two groups having(not necessarily genuine) isolated orderings, we construct a new group having anisolated left ordering by using amalgamated free product over Z .In almost all cases, the obtained isolated orderings are genuine. Our constructioncan be seen as an extension of (2) of known examples, but it is completely differentfrom the Dehornoy-like orderings construction. In fact, we will see that some of theso far orderings cannot be obtained from Dehornoy-like orderings.The following is the summary of main result of this paper. Recall that for g ∈ G and a left ordering < G of G , < G called a g -right invariant ordering if the ordering < G is preserved by the right multiplication of g , that is, a < G b implies ag < G bg for all a, b ∈ G . Theorem 1.1 (Construction of isolated left ordering via partially central cyclicamalgamation) . Let G and H be finitely generated groups. Let z G be a non-trivialcentral element of G , and z H be a non-trivial element of H .Let G = { g , . . . , g m } be a finite generating set of G which defines an isolatedleft ordering < G of G . We take a numbering of elements of G so that < G g < G · · · < G g m holds. Similarly, let H = { h , . . . , h n } be a finite generating set of H which defines an isolated left ordering < H of H such that the inequalities < H h < H · · · < H h n hold.We assume the cofinality assumptions [CF(G)] , [CF(H)] , and the invarianceassumption [INV(H)] . [ CF ( G )] g i < G z G holds for all i. [ CF ( H )] h i < H z H holds for all i. [ INV ( H )] < H is a z H -right invariant ordering . Let X = G ∗ Z H = G ∗ h z G = z H i H be an amalgamated free product of G and H over Z . For i = 1 , . . . , m , let x i = g i z − H h . Then we have the following results: (i) The generating set { x , . . . , x m , h , . . . , h n } of X defines an isolated leftordering < X of X . (ii) The isolated ordering < X does not depend on the choice of the generatingsets G and H . Thus, < X only depends on the isolated orderings < G , < H and the elements z G , z H . (iii) The natural inclusions ι G : G → X and ι H : H → X are order-preservinghomomorphisms. (iv) 1 < X x < X · · · < X x m < X h < X · · · < X h n < X z H = z G . Moreover, z = z G = z H is < X -positive cofinal and the isolated ordering < X is a z -right invariant ordering. (v) r ( < X ) ≤ r ( < G ) + r ( < H ) . TETSUYA ITO (vi)
Let Y be a non-trivial proper subgroup of X . If Y is < X -convex, then Y = h x i , the infinite cyclic group generated by x . We call the construction of isolated ordering described in Theorem 1.1 the par-tially central cyclic amalgamation construction .As we will see in Lemma 2.3 in Section 2.1, the cofinality assumption
CF(G) (resp.
CF(H) ) is understood as an assumption on z G and < G (resp. z H and < H ). Thus, Theorem 1.1 (ii) shows that the choice of the generating sets G and H is not important, though it is useful to describe and understand the isolatedordering < X . The generating sets G and H play rather auxiliary roles and are notessential in our partially central cyclic amalgamation construction. This makes asharp contrast with the construction using Dehornoy-like orderings, since in theDehornoy-like ordering construction we need to use a special generating set derivedfrom Dehornoy-like ordering having the nice property which we called the PropertyF. Theorem 1.1 (iii) shows that the partially central cyclic amalgamation construc-tion can be seen as a mixing of two isolated orderings < G and < H . We remarkthat Theorem 1.1 (iv) ensures that we can iterate the partially central cyclic amal-gamation construction. Thus, we can actually produce many isolated orderings byusing the partially central cyclic amalgamation constructions from known isolatedorderings.The proof of Theorem 1.1 (i) is constructive and will actually provide an algo-rithm to determine the isolated ordering < X . In particular, the isolated ordering < X can be determined algorithmically if we have algorithms to compute the isolatedorderings < G and < H , as we will see in Section 2.7.It is interesting to compare our amalgamation with other natural operationson groups. Unlike the partial central cyclic amalgamation (the amalgamated freeproduct over Z , used in Theorem 1.1), the usual free product does not preservethe property that the group has an isolated left ordering. Here is the simplestcounter-example: the free group of rank two F = Z ∗ Z has no isolated orderings[7], whereas the infinite cyclic group Z has (non-genuine) isolated orderings, since itadmits only two left orderings. Indeed, recently Rivas [9] proved that free productsof groups do not have any isolated left orderings. Similarly, the direct productsof groups also do not preserve the property that the group has an isolated leftordering: the free abelian group of rank two Z × Z has no isolated orderings [10].The plan of this paper is as follows: In Section 2 we prove Theorem 1.1. Themain technical tool of the proof is a reduced standard factorization , which servesas some kind of normal form of elements in X , adapted to the generating set { x , . . . , x m , h , . . . , h n } . In Section 3 we give some examples of isolated orderingsobtained by applying Theorem 1.1. We observe that our examples have variousinteresting properties, which have not been appeared for known examples.2. Construction of isolated left orderings
Let S = { s , . . . , s n } be a finite generating set of G and let S − = { s − , . . . , s − n } .We denote by S ∗ the free semigroup generated by S . That is, S ∗ is a set of non-empty words on S . We say an element of S ∗ (resp. ( S − ) ∗ ) an S -positive word (resp. an S -negative word ). We will often use a symbol P ( S ) (resp. N ( S )) torepresent an S -positive (resp. S -negative) word. ONSTRUCTION OF ISOLATED LEFT ORDERINGS 5
Cofinality and Invariance assumptions.
First of all we review the assump-tions in the statement of Theorem 1.1 again, and deduce their direct consequences.This clarifies the role of each hypothesis in Theorem 1.1.Let G and H be finitely generated groups having an isolated left ordering < G and < H respectively. Let z G ∈ G be a non-trivial central element of G , and let z H be a non-trivial element of H , which might be noncentral. We consider the group X obtained as an amalgamated free product over Z X = G ∗ Z H = G ∗ h z G = z H i H. Let G = { g , . . . , g m } be a generating set of G which defines an isolated leftordering < G of G . We take a numbering of elements of G so that 1 < G g < G · · · < G g m holds. Similarly, let H = { h , . . . , h n } be a generating set of H whichdefines an isolated left ordering < H of H , and we assume that the inequalities1 < H h < H · · · < H h n hold.Recall that an element g ∈ G is called the < G -minimal positive element if g isthe < G -minimal element in the positive cone P ( < G ). In other words, the inequality1 < G g ′ ≤ G g implies g = g ′ . A left ordering < G is called discrete if < G has a < G -minimal positive element. Otherwise, < G is called dense .As the next lemma shows, the choice of the numbering of G (resp. H ) impliesthat g (resp. h ) is the < G -minimal (resp. < H -minimal) positive element. Inparticular, g (resp. h ) is independent of the choice of the generating set G (resp. H ). Lemma 2.1.
Let G = { g , . . . , g m } be a generating set of a group G which definesan isolated left ordering < G of G . Assume that g is the < G -minimal element in theset G . Then g is the < G -minimal positive element. In particular, < G is discrete.Moreover, < G is a g -right invariant ordering.Proof. Assume g ∈ G satisfies the inequalities 1 < G g ≤ G g .1 < G g means that g is written as a G -positive word g = g i · · · g i l . Then g − g = ( g − g i ) g i · · · g i l ≤ G i = 1 and l = 1, that is, g = g .The g -right invariance of the ordering < G now follows from the fact that g isthe < G -minimal positive element: If a < G b , then 1 < G a − b < G a − bg . Thus, g < G a − bg so ag < G bg . (cid:3) To obtain an isolated ordering of X from < G and < H , we impose the follow-ing assumptions, which we call the cofinality assumption for G and H , and the invariance assumption .[ CF ( G )] g i < G z G holds for all i. [ CF ( H )] h i < H z H holds for all i. [ INV ( H )] < H is a z H -right invariant ordering . Here we remark that the invariance assumption for < G is automatically satisfied:that is, < G is a z G -right invariant ordering since we have chosen z G so that it is acentral element.First we observe the following simple lemma. TETSUYA ITO
Lemma 2.2.
Let < H be a discrete left ordering of a group H , and let h be the < H -minimal positive element. If < H is an h -right invariant ordering for h ∈ H ,then h commutes with h .Proof. < H is an h -right invariant ordering, so hh h − > H h − h h > H h is the < H -minimal positive element, so hh h − ≥ H h and h − h h ≥ H h . Thus,we get hh ≥ H h h and h h ≥ H hh , hence hh = h h . (cid:3) By Lemma 2.1 and Lemma 2.2, the invariance assumption [INV(H)] impliesthat z H commutes with h .For a left-ordering < G of G , an element g ∈ G is called < G -cofinal if for all g ′ ∈ G , there exist integers m and M such that g m < G g ′ < G g M holds. Althoughthe cofinality assumptions [CF(G)] and [CF(G)] involve the generating sets G and H , if we assume the invariance assumption [INV(H)] then these assumptionsshould be regarded as assumptions on z G , z H and the isolated orderings < G , < H as the next lemma shows. Lemma 2.3.
Assume the invariance assumption [INV(H)] is satisfied. A gener-ating set H satisfying the cofinality assumption [CF(H)] exists if and only if z H is < H -positive cofinal and H = h z H i . Here h z H i represents the subgroup of H gener-ated by z H . Moreover, in such case we may choose a generating set H so that thecardinal of H is equal to the rank of the isolated ordering < H .Proof. In the following, we assume the invariance assumption [INV(H)] .Assume that a generating set H satisfies the cofinality assumption [CF(H)] .Then by the invariance assumption [INV(H)] , z H is < H -positive cofinal and H = h z H i .We show the converse: if z H is < H -positive cofinal and H = h z H i , then we canchoose a generating set H = { h , . . . , h k } so that H defines the isolated ordering < H , and that H satisfies [CF(H)] . Moreover, we will show that we can choose k ,the cardinal of H , so that k is equal to r ( < H ).Let us take a generating set H ′ = { h ′ , . . . , h ′ k } of H which defines the isolatedordering < H . By definition of rank, we may choose H ′ so that k = r ( < H ) holds.With no loss of generality, we may assume that h ′ < H · · · < H h ′ s ≤ H z H < H h ′ s +1 < H · · · < H h ′ k . Since z H is < H -cofinal, for each i there is a non-negative integer N i such that1 < H z − N i H h ′ i ≤ z H . Let us put h i = z − N i H h ′ i . By assumption, h ′ i = h i if i ≤ s .By the hypothesis H = h z H i , we have a strict inequality z H > H h ′ = h . Thusif necessary, by replacing h i with h − h i , we may assume that h i = z H for all i .We show that z H is written as an { h ′ , . . . , h ′ s } -positive word. Assume that z H = V h ′ i W , where i > s and V, W are H ′ -positive or non-empty words. Then z H W − = V h ′ i > H V z H , hence we get 1 ≥ H W − > H z − H V z H . However, < H is a z H -right invariant ordering, hence z − H V z H ≥ H
1. This is a contradiction.Therefore, the generating set H = { h , . . . , h k } also defines the isolated ordering < H . By construction, H is a generating set which satisfies the cofinality assumption [CF(H)] with cardinal k = r ( < G ). (cid:3) Thus, under the invariance assumption [INV(H)] , we can always find a gener-ating set H which defines < H and satisfies the cofinality assumption [CF(H)] , if ONSTRUCTION OF ISOLATED LEFT ORDERINGS 7 the conditions on < H and z H in Lemma 2.3 are satisfied. Moreover, if necessarywe may choose H so that the cardinal of H is equal to the rank of < H .Since for z G and < G , the invariance assumption is automatically satisfied, wecan always find a generating set G which defines < G and satisfies the cofinalityassumption [CF(G)] if z G is < G -positive cofinal and G = h z G i .Now we put ∆ H = z H h − . Since z H and h do not depend on the choice of thegenerating set H , the same holds for ∆ H . As an element of H , ∆ H is characterizedby the following property. Lemma 2.4. ∆ H is the < H -maximal element which is strictly smaller than z H .Proof. Assume that z H h − = ∆ H ≤ H h < H z H holds for some h ∈ H . Then h − ≤ H z − H h < H
1. By Lemma 2.1, h − is the < H -maximal element which isstrictly smaller than 1, so z − H h = h − . Hence h = z H h − . (cid:3) Finally, we put x i = g i ∆ − H = g i z − H h and let X = { x , . . . , x m } . Then {X , H} generates the group X . The following lemma is rather obvious, but plays an im-portant role in the proof of Theorem 1.1. Lemma 2.5. z H = z G commutes with all x i .Proof. By Lemma 2.2, z H commutes with ∆ H = z H h − . Since z H = z G commuteswith all g i , we conclude that z H commutes with all x i = g i ∆ − H . (cid:3) Property A and Property C criteria.
To prove that {X , H} defines anisolated left ordering < X of X , we use the following criterion which was used in thetheory of the Dehornoy ordering of the braid groups [1] and Dehornoy-like orderings[3, 8]. Here we give the most general form of this kind of arguments. Definition 2.6.
Let S = { s , . . . , s m } be a generating set of a group G and let W be a sub-semigroup of ( S ∪ S − ) ∗ .(1) We say W has the Property A (Acyclic Property) if no word in W representthe trivial element of G .(2) We say W has the Property C (Comparison Property) if for each non-trivialelement g ∈ G , either g or g − is represented by a word w ∈ W . Proposition 2.7.
Let W be a sub-semigroup of ( S ∪ S − ) ∗ . Let P = π ( W ) , where π : ( S ∪ S − ) ∗ → G is the natural projection. Then P is equal to a positive cone ofa left ordering of G if and only if W has Properties A and C.Proof. If W is a positive cone of a left ordering, then it is obvious that W hasProperties A and C. We show the converse. Since W is a sub-semigroup, P is asub-semigroup of G . By Property C, G = P ∪ { } ∪ P − . Property A implies that1 P , hence G is decomposed as a disjoint union G = P ⊔ { } ⊔ P − . This showsthat P is a positive cone of a left ordering. (cid:3) Definition 2.8.
The set of words W in Proposition 2.7 is called the languagedefining the corresponding left-ordering.It is an interesting problem to ask whether or not one can choose a languagedefining an arbitrary left-ordering < G so that it is a regular language over finitealphabets: This is related to order-decision problems which we will consider inSection 2.7, but in this paper we will not treat this problem.As a special case, we get a criterion for a finite generating set to define an isolatedordering, which will be used to show {X , H} indeed defines an isolated ordering. TETSUYA ITO
Corollary 2.9.
A finite generating set G = { g , . . . , g m } of a group G defines anisolated ordering of G if and only if the following conditions [Property A] and [Property C] hold: Property A: If g ∈ G is represented by a G -positive word, then g = 1 . Property C: If g = 1 , then g is represented by either a G -positive or a G -negative word. Reduced standard factorization.
Now we start to show that {X , H} indeeddefines an isolated left ordering of X . From now on, we take G, H, X, < G , < H , G , H , X as in assumptions in Theorem 1.1, and we always assume the cofinalityassumptions [CF(G)] , [CF(H)] , and the invariance assumption [INV(H)] .As the first step of the proof, we introduce a notion of reduced standard factoriza-tion, which serves as a certain kind of normal form of X adapted to the generatingset {X , H} .Let P X be the sub-semigroup of X generated by X = { x , . . . , x m } . A standardfactorization of x ∈ X is a factorization of x ∈ X of the form F ( x ) = rp q · · · p l q l where r, q , . . . , q l ∈ H , p , . . . , p l ∈ P X satisfy the conditions(1) q i > H i = l ), and q l ≥ H
1, and(2) q i = z NH for all N > x ∈ X admits a standard factorization. Recall that x − i = ∆ H g − i = ∆ H z − H ( z G g − i ). Since 1 < G ( z G g − i ), ( z G g − i ) is written as a G -positive word. By rewriting such a G -positive word as an {X , H} -positive word,we may write x − i as x − i = z − H ∆ H P i ( X , H )where P i ( X , H ) denotes a certain {X , H} -positive word. Thus, we can write x without using { x − , . . . , x − m } .Moreover, z H is < H -cofinal so for h ∈ H there exists N ∈ Z such that z NH h > H < H is a z H -right invariant ordering, and that z H commutes with all X (Lemma 2.5). Thus, the above consideration shows that weare able to get a standard factorization F ( x ) = rp q · · · p l q l . The complexity of a standard factorization F ( x ) = rp q · · · p l q l is defined to be l , and denoted by c ( F ).A distinguished subfactorization of a standard factorization F ( x ) is, roughlysaying, a part of the standard factorization F ( x ) which can be regarded as a G -positive word, defined as follows.We say a subfactorization(1) w = ( q i p i +1 q i +1 · · · p i + r q i + r )in a standard factorization F ( x ) is a distinguished subfactorization if it satisfies thefollowing two conditions:(1) q j = ∆ H for all j = i, i + 1 , . . . , i + r .(2) p j ∈ X for all j = i + 1 , . . . , i + r . ONSTRUCTION OF ISOLATED LEFT ORDERINGS 9
That is, a distinguished subfactorization is a part of standard factorization whichis written as(2) w = ∆ H x j i +1 ∆ H x j i +2 · · · x j i + r ∆ H We will express the distinguished subfactorization w (2) by using a G -positiveword g w as follows: Let us take x a ∈ X so that p ′ i = p i x − a ∈ P X ∪ { } (such achoice of x a might be not unique), and write a standard factorization F ( x ) as F ( x ) = rp q · · · p l q l = rp q · · · p i − q i − ( p i x − a )( x a q i · · · p i + r q i + r ) p i + r +1 q i + r +1 · · · p l q l = rp q · · · p i − q i − p ′ i ( x a ∆ H x j i +1 ∆ H x j i +2 · · · x j i + r ∆ H ) p i + r +1 q i + r +1 · · · p l q l . Let us put g w = g j i +1 · · · g j + i + r . We call g w the corresponding G -positive word (element) of the distinguished sub-factorization w . Since g i = x i ∆ H , x a ∆ H x j i +1 ∆ H x j i +2 · · · x j i + r ∆ H = g a g j i +1 · · · g j i + r = g a g w . Thus, if w is a distinguished subfactorization in F ( x ), by choosing x a we mayexpress x as x = rp q · · · p i − q i − p ′ i [ g a g w ] p i + r +1 q i + r +1 · · · p l q l . by using the corresponding G -positive word g w .Next we introduce a notion of reducible distinguished subfactorization . Let w bea distinguished subfactorization of F ( x ) as taken in (1). Let us take x u ∈ X sothat p ′ i + r +1 = x − u p i + r +1 ∈ P X ∪ { } . As for the choice of x a above, such x u maynot unique. If such x u does not exist, that is, p i + r +1 = 1, we take x u = 1. We saya distinguished subfactorization w is reducible if for any choice of such x a and x u ,we have an inequality g a g w g u ≥ G z G . Otherwise, that is, if one can choose x a and x u so that g a g w g u < G z G holds, then we say w is irreducible .Now we define the notion of a reduced standard factorization , which plays animportant role in the proof of both Property A and Property C. Definition 2.10 (Reduced standard factorization) . Let F ( x ) = rp q · · · p l q l be astandard factorization. We say F is reduced if q i < H z H for all i and F containsno reducible distinguished subfactorization.We say a distinguished subfactorization w of a standard factorization F is maxi-mal if there is no other distinguished subfactorization w ′ of F whose corresponding G -positive word g w ′ contains g w as its subword. For any < G -positive elements g, g ′ , g ′′ , since z G is central, if g ≥ G z G then g ′ gg ′′ ≥ G z G . Thus, to see whether astandard factorization is reducible or not, it is sufficient to check that all maximaldistinguished subfactorization are irreducible. Example 2.11.
A distinguished subfactorization and related notions are slightlycomplex, so here we give an example. Let us consider the case X = { x , x } andtake a standard factorization of the form(3) F ( x ) = ( x x )∆ H x ∆ H x ∆ H ( x x ) h , for example. In the standard factorization (3) w = ∆ H x ∆ H is a distinguished subfactoriza-tion. The corresponding G -positive word is g w = g . In this case, we may choose x a = x since ( x x ) x − = x ∈ P X . So we are able to write x as x = x [ g g ] x ∆ H ( x x ) h . However, the distinguished sub-factorization w is not maximal: it is included inanother distinguished subfactorization w ′ = ∆ H x ∆ H x ∆ H , and we may write x = x [ g ( g g )]( x x ) h . The distinguished subfactorization w ′ is maximal.Is w ′ reducible ? To see this, first we need to determine all possibilities of x a and x u in the definition of reducible distinguished subfactorization. Assume that( x x ) x − ∈ P X ∪ { } , but x − ( x x ) P X ∪ { } . Then x a = x or x , and x u = x . Hence our definition says, w ′ is reducible if and only if g ( g g ) g ≥ G z G , and g ( g g ) g ≥ G z G hold.First we show the existence of the reduced standard subfactorization. The proofof the next lemma utilizes the standard form of amalgamated free products, andmainly works in the generating set {G , H} . Lemma 2.12.
Every element x ∈ X admits a reduced standard subfactorization.Proof. Since X is an amalgamated free product of G and H , every x ∈ X is writtenas x = q f q f q · · · f l q l where q i ∈ H , f i ∈ G , and q i = z NH and f i = z NG for any N ∈ Z and i > z G is < G -cofinal, for each i > N i ∈ Z which satisfies z N i G < G f i < G z N i +1 G . We put f ∗ i = z − N i G f i . Then f ∗ i satisfies the inequality1 < G f ∗ i < G z G Similarly, since z H is < H -cofinal, for each i > M i which satisfies theinequality z M i H ≤ H ∆ H q i < H z M i +1 H . Let L i = P j>i ( N j + M j ), and put q ∗ i = z − L i H ( z − M i H ∆ H q i ) z L i H . Since < H is a z H -right invariant ordering, 1 ≤ H q ∗ i < H z H holds. We have assumed that q i = z NH ,so we have q ∗ i = ∆ H . Thus, 1 ≤ H q ∗ i < H ∆ H . ONSTRUCTION OF ISOLATED LEFT ORDERINGS 11
Then we get a reduced standard factorization of x as follows. First we modifythe first expression of x as x = q f q · · · f l q l = q ( z N G f ∗ ) q ( z N G f ∗ ) · · · ( z N l − G f ∗ l − ) q l − ( z N l G f ∗ l ) q l = ( q z N H ) f ∗ ( q z N H ) · · · f ∗ l − ( q l − z N l H ) f ∗ l q l = ( q z N H ) f ∗ ( q z N H ) · · · f ∗ l − ( q l − z N l H ) f ∗ l ∆ − H z M l H ( z − M l H ∆ H q l )= ( q z N H ) f ∗ ( q z N H ) · · · f ∗ l − ( q l − z N l H z M l H )( f ∗ l ∆ − H ) q ∗ l = ( q z N H ) f ∗ ( q z N H ) · · · f ∗ l − ∆ − H z N l + M l H ( z − N l − M l H ∆ H q l − z N l + M l H )( f ∗ l ∆ − H ) q ∗ l = ( q z N H ) f ∗ ( q z N H ) · · · z N l + M l H ( f ∗ l − ∆ − H ) q ∗ l − ( f ∗ l ∆ − H ) q ∗ l = · · · = ( q z L H )( f ∗ ∆ − H ) q ∗ · · · ( f ∗ l − ∆ − H ) q ∗ l − ( f ∗ l ∆ − H ) q ∗ l . Now let us write f ∗ i = P i ( G ) g k i , where P i ( G ) is a G -positive or an empty word.Since g i = x i ∆ H , we may express P i ( G ) as an {X , H} -positive (or, empty) word.Hence by rewriting each P i ( G ) as an {X , H} -positive (or, empty) word, we get astandard factorization F ( x ) = ( q z L H )[ P ( G )] x k q ∗ · · · [ P l − ( G )] x k l − q ∗ l − [ P l ( G )] x k l q ∗ l . Recall that q ∗ i = ∆ H for all i . Thus for a maximal distinguished subfactorization w in F ( x ), we may choose g a and g u so that g a g w g u = P i ( G ) g k i holds for some i .Since P i ( G ) g k i = f ∗ i < G z G , this implies that all distinguished sub-factorizationsare irreducible. Hence F ( x ) is a reduced standard factorization. (cid:3) Reducing operation and the proof of Property A.
In the proof ofLemma 2.12 given in previous section, we mainly used the generating set {G , H} .In this section we give an alternative way to get a reduced standard factorization,which works mainly in the generating set {X , H} .We say a standard factorization F ( x ) = rp q · · · p l q l is pre-reduced if 1 < H q i < H z H holds for all i . It is rather easy to see pre-reduced standard factorizationexists. Lemma 2.13 (Existence of pre-reduced standard factorization) . Every element x ∈ X admits a pre-reduced standard factorization.Proof. Let F ( x ) = rp q · · · p l q l be a standard factorization. For each i , take M i ≥ z M i H < H q i < H z M i +1 H . Let L i = P j ≥ i M i and q ∗ i = z − L i H q i z L i +1 H = z − L i +1 H ( z − M i H q i ) z L i +1 H . Since < H is z H -right invariant, 1 < H q ∗ i < H z H . Therefore,we get a pre-reduced standard factorization x = rp q · · · p l q l = rp q · · · p l − q l − p l ( z M l H q ∗ l )= rp q · · · p l − ( q l − z M l H ) p l q ∗ l = rp q · · · p l − z − M l − M l − H ( z − M l − M l − H q l − z M l H ) p l q ∗ l = rp q · · · p l − z − L l − H q ∗ l − p l q ∗ l = · · · = ( rz − L H ) p q ∗ · · · p l q ∗ l . (cid:3) To show we are actually able to get reduced standard factorization, we observethat we are able to eliminate all reducible distinguished subfactorizations. Let d ( F ) be the number of maximal reducible distinguished subfactorizations. Thenext lemma gives alternative proof that a reduced standard factorization exists. Itsays that by induction on ( d ( F ) , c ( F )) for pre-reduced factorization F , we are ableto get reduced standard factorization. Lemma 2.14 (Reducing operation) . Let F ( x ) = rp q · · · p l q l be a pre-reducedstandard factorization of x ∈ X . If F ( x ) contains a reducible distinguished sub-factorization, then we can find another pre-reduced standard factorization F ′ ( x ) = r ′ p ′ q ′ · · · which satisfies d ( F ′ ) < d ( F ) or, d ( F ′ ) = d ( F ) and c ( F ′ ) < c ( F ) . More-over, if r > H then r ′ > H .Proof. Let w = q i p i +1 · · · p s − q s − be a reducible maximal distinguished subfactor-ization in F ( x ). Thus, we may assume that the pre-reduced standard factorization F ( x ) is written as F ( x ) = rp q · · · p i − q i − p ′ i [ g a g w ] x u p ′ s q s · · · p l q l where(1) p ′ i = p i x − a and p ′ s = x − u p s (2) p ′ i , p ′ s ∈ P X ∪ { } .(3) g a g w g u ≥ G z G .Now take N > z NG < G g a g w g u ≤ G z N +1 G , and for j < i let q ∗ j = z − NH q j z NH . Then we may write x as x = rp q · · · p i − q i − p ′ i [ g a g w ] x u p ′ s q s · · · p l q l = rp q · · · p i − q i − p ′ i z NG ( z − NG g a g w g u )∆ − H p ′ s q s · · · p l q l = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i ( z − NG g a g w g u )∆ − H p ′ s q s · · · p l q l . First assume that ( z − NG g a g w g u ) = z G = z H . Then we write x as x = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i ( z H ∆ − H ) p ′ s q s · · · p l q l = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i h p ′ s q s · · · p l q l If p ′ i = 1 and p ′ s = 1, then we get a pre-reduced standard factorization(4) F ′ ( x ) = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i h p ′ s q s · · · p l q l . In F ′ ( x ), we removed the reducible distinguished subfactorization w and no distin-guished subfactorization is created, so d ( F ′ ) < d ( F ).If p ′ i = 1 or p ′ s = 1, then the standard factorization (4) might fail to be pre-reduced. We construct a pre-reduced standard factorization F ′′ from the standardfactorization (4) by using the argument of proof of Lemma 2.13. In such case,we might produce one new reducible maximal distinguished subfactorization, so ingeneral d ( F ′′ ) ≤ d ( F ) although we have removed w from F ( x ). In this case wehave c ( F ′′ ) < c ( F ).(Here is a simple example where d ( F ′′ ) does not decrease: assume that p ′ i = 1, p ′ s ∈ X , and q ∗ i − h = ∆ H , and that w = q s · · · is a distinguished subfactorization:then we get a new maximal distinguished subfactorization w = · · · ( q ∗ i − h ) p ′ s q s · · · ONSTRUCTION OF ISOLATED LEFT ORDERINGS 13 in F ′ ( x ). This maximal distinguished subfactorization might be reducible, so d ( F ′′ ) = d ( F ) may occur.)Next assume that ( z − NG g a g w g u ) = z G . Let us put g ′ = ( z − NG g a g w g u ) g − andwrite x as F ( x ) = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i ( z − NG g a g w g u )∆ − H p ′ s q s · · · p l q l = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i g ′ g ∆ − H p ′ s q s · · · p l q l = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i g ′ ( x p ′ s ) q s · · · p l q l . If g ′ = 1, then we get a pre-reduced standard factorization F ′ ( x ) = ( rz NH ) p q ∗ · · · p i − q ∗ i − ( p ′ i x p ′ s ) q s · · · p l q l . such that d ( F ′ ) < d ( F ).If g ′ > G
1, then let us write g ′ = g a ′ P ( G ) where P ( G ) is a G -positive, or emptyword. By rewriting P ( G ) as an {X , H} -positive word, we get a new pre-reducedstandard factorization F ′ ( x ) = ( rz NH ) p q ∗ · · · p i − q ∗ i − p ′ i [ g a ′ P ( G )]( x p ′ s ) q s · · · p l q l . Observe that P ( G ) gives rise to a maximal distinguished subfactorization w ′ in F ′ ( x ) such that g w ′ = P ( G ). By Lemma 2.1, < G is a g -right invariant ordering,so 1 ≤ G g ′ < G z G g − , so g a ′ g w ′ g < G z G . Hence the maximal distinguishedsubfactorization w ′ in F ′ ( x ) is irreducible. By construction, all other maximalreducible distinguished subfactorizations in F ′ ( x ) are derived from the pre-reducedfactorization F ( x ). Since we have removed the maximal reducible distinguishedsubfactorization w in F ( x ), so d ( F ′ ) < d ( F ).Moreover, by construction we have always r ≤ H r ′ . In particular, 1 < H r ′ if1 < H r , (cid:3) Now we are ready to prove Property A.
Proposition 2.15 (Property A) . If x is expressed as an {X , H} -positive word,then x = 1 .Proof. Assume that x is expressed by an {X , H} -positive word. Such a word expres-sion can be modified to a standard factorization which is also an {X , H} -positiveword: By the proof of Lemma 2.13, we can modify such a standard factorizationso that it is pre-reduced, preserving the property that it is also an {X , H} -positiveword. By Lemma 2.14, we may modify the {X , H} -positive pre-reduced standardexpression F ( x ) so that it is an {X , H} -positive reduced standard factorization.Now let us rewrite F ( x ) as a word on {G , H} as follows. Let w be a maximaldistinguished subfactorization in F ( x ) so we may write F ( x ) as F ( x ) = rp q · · · p i − q i − p ′ i [ g a g w ] p s q s · · · p l q l Since w is irreducible, we may choose g a and x u ∈ X so that p ′ s = x − u p s , p ′ i ∈ P X ∪ { } and that g a g w g u < G z G . Then we write x as F ( x ) = rp q · · · p i − q i − p ′ i ( g a g w g u )∆ − H p ′ s q s · · · p l q l , and regard ( g a g w g u ) as a G -positive word.Iterating this rewriting procedure for each maximal distinguished subword, andrewriting the rest of x i in F ( x ) as a word on {G , H} by using the relation x i = g i ∆ − H , we finally write x as x = W V W · · · V n W n where W i is a word on H ± and V i is a word on G ± . By construction, V i ∈ X or V i = g a g w g u where g w is a maximal distinguished subfactorization in F ( x ). Sincewe have chosen g a g w g u < G z G , this implies that, V i
6∈ h z G i for all i . Similarly, theassumption that F ( x ) is reduced implies that we may choose W i
6∈ h z H i for i > x = 1, since X = G ∗ h z G = z H i H . (cid:3) Proof of Property C.
Next we give a proof of Property C. To begin with,we observe a simple, but useful observation.
Lemma 2.16. h − j x i = N ( X , H )∆ − H where N ( X , H ) represents an {X , H} -negative word.Proof. Since z H = z G and x i = g i ∆ − H , we have z H = g i g − i z G g − g = x i ∆ H ( g − i z G g − ) x ∆ H . Therefore h − j x i = ( h − j z H ∆ − H ) x − ( z − G g g i )∆ − H = ( h − j h ) x − ( z − G g g i )∆ − H . Since z − G g i < G g is the < G -minimal positive element, z − G g i ≤ G g − . Hence z − G g g i ≤ G
1. Thus, ( h − j h ) x − ( z − G g g i ) is written as an {X , H} -negative word. (cid:3) Now we are ready to prove Property C . Proposition 2.17 (Property C) . Each non-trivial element x ∈ X is expressed byan {X , H} -positive word or an {X , H} -negative word.Proof. Let x be a non-trivial element of X and take a reduced standard factorizationof x , F ( x ) = rp q · · · p l q l . If r ≥ H r can be written as an H -positive or empty word, hence we may express x as an {X , H} -positive word.By induction on l = c ( F ), we prove that x is expressed by an {X , H} -negativeword under the assumption that r < H q = ∆ H . Since r < H
1, we can express r as r = N ( H ) h − ,where N ( H ) is an H -negative word or an empty word. Take an X -positive wordexpression of p = x i x i · · · x i p . Then by Lemma 2.16, rp p q · · · = ( N ( H ) h − )( x i x i · · · x i p ) q p q · · · = N ( H )( h − x i ) x i · · · x i p q p q · · · = N ( X , H )∆ − H x i · · · x i p q p q · · · = N ( X , H )( h − x i ) · · · x i p q p q · · · = · · · = N ( X , H )∆ − H q p q · · · .N ( X , H ) represents an {X , H} -negative word.Since F ( x ) is a reduced standard factorization, q < H z H . By Lemma 2.4 ∆ H is the < H -maximal element of H which is strictly smaller than z H , so q ≤ H ∆ H . We have assumed that q = ∆ H so (∆ − H q ) < H
1. Thus the subword
ONSTRUCTION OF ISOLATED LEFT ORDERINGS 15 (∆ − H q ) p q · · · p l q l is a reduced standard factorization with complexity ( l − − H q ) p q · · · p l q l is written as an {X , H} -negative word, hence weconclude that x is written as an {X , H} -negative word.Next assume that q = ∆ H . Let w = q p q · · · p s − q s − be a maximal distin-guished subfactorization of F ( x ) which contains q . Thus, the reduced standardfactorization S is written as F ( x ) = rp ′ [ g a g w ] x u p ′ s q s p s +1 · · · p l q l where p ′ = p x − a , p ′ s = x − u p s ∈ P X ∪ { } .Then by Lemma 2.16, x = rp ′ [ g a g w ] x u p ′ s q s p s +1 · · · p l q l = N ( X , H ) h − [ g a g w ] x u ∆ H ∆ − H p ′ s q s · · · p l q l = N ( X , H ) h − [ g a g w g u ]∆ − H p ′ s q s · · · p l q l = N ( X , H )∆ H ( z − G g a g w g u )∆ − H p ′ s q s · · · p l q l The distinguished subfactorization w is irreducible so we may choose x u and g a so that z − G g a g w g u < G z − G g a g w g u is written as a G -negative word. By expressing a G -negative word expression of z − G g a g w g u as an {X , H} -negative word, we conclude that z − G g a g w g u is written as a word of the form∆ − H N ( X , H ). Hence x = N ( X , H )∆ H ∆ − H N ( X , H )(∆ − H p ′ s q s · · · p l q l )= N ( X , H )(∆ − H p ′ s q s · · · p l q l ) . If p ′ s = 1, then (∆ − H p ′ s q s · · · p l q l ) is a reduced standard factorization having thecomplexity less than l . Hence by induction, (∆ − H p ′ s q s · · · p l q l ) is expressed by an {X , H} -negative word.If p ′ s = 1, then q s = ∆ H since w was a maximal distinguished subfactorization.Hence q s < H ∆ H , and (∆ − H q s ) p s +1 · · · p l q l is a reduced standard factorizationwith complexity less than l . By induction, (∆ − H q s ) p s +1 · · · p l q l is expressed by an {X , H} -negative word.Thus in either case, we conclude x is expressed by an {X , H} -negative word. (cid:3) Proof of Theorem 1.1.
Now we are ready to prove our main theorem. of Theorem 1.1. (i): In Proposition 2.15 and Proposition 2.17, we have alreadyconfirmed the Properties A and C for the generating set {X , H} . By Corollary 2.9the generating set {X , H} indeed defines an isolated left ordering < X of X .(ii): Let G ′ = { g ′ , . . . } and H ′ = { h ′ , . . . } be other generating sets of G and H satisfying [CF(G)] and [CF(H)] . Recall that ∆ H = z H h − does not depend onthe choice of a generating set H . Let x i = g i ∆ − H , x ′ i = g ′ i ∆ − H , X = { x , . . . , } , and X ′ = { x ′ , . . . , } .Since H and H ′ are generators of the same semigroup, we may write h i as an H ′ -positive word. Similarly, since G and G ′ are generators of the same semigroup,we may write g i as a G ′ -positive word g i = g ′ i g ′ i · · · g ′ i l . Thus, x i = g i ∆ − H = g ′ i g ′ i · · · g ′ i l ∆ − H = x ′ i ∆ H x ′ i ∆ H · · · x ′ i l − ∆ H x ′ i l so x i is written as an {X ′ , H ′ } -positive word. Thus, if x ∈ X is expressed byan {X , H} -positive word, then x is also represented by an {X ′ , H ′ } -positive word.By interchanging the roles of {G , H} and {G ′ , H ′ } , we conclude that {X , H} and {X ′ , H ′ } generates the same sub-semigroup of X so they define the same isolatedordering of X .(iii): This is obvious from the definition of < X .(iv): The inequality h < X h < X · · · < X h n follows from the definition of < X . By Lemma 2.16, x i < X h for all i . Now we show x i < X x j if i < j .Since g i < G g j if i < j , g − i g j is written as a G -positive word. Now by definition g i = x i ∆ H , so we may express a G -positive word expression of g − i g j as an {X , H} -positive word expression of the form P i,j ( X , H )∆ H , where P i,j ( X , H ) representsan {X , H} -positive word. Therefore x − i x j = ∆ H g − i g j ∆ − H = ∆ H P i,j ( X , H ), so x i < X x j . The assertion that z = z G = z H is < X -positive cofinal is obvious. Tosee that < X is a z -right invariant ordering, we observe that z − x i z = x i > X z − h j z > X
1. Now for x, x ′ ∈ X , assume x < X x ′ , so x − x ′ is writ-ten as {X , H} -positive word w = s · · · s m , where s i denotes x j or h j . Then z − ( x − x ′ ) z = ( z − s z ) · · · ( z − s m z ) > X
1, hence xz < X x ′ z .(v): Recall that by Lemma 2.3, we may choose the generating sets G and H so that the cardinal of G , H are equal to r ( < G ), r ( < H ) respectively. Thus, r ( < X ) ≤ r ( < G ) + r ( < H ).(vi): We prove that h x i is the unique < X -convex non-trivial proper subgroupof X . Recall by (2), (4) and Lemma 2.1, x is the minimal < X -positive elementof X , hence x does not depend on a choice of G and H . In particular, h x i is anon-trivial < X -convex subgroup.Let C be a < X -convex subgroup of X . Assume that C ⊃ h x i . Let y ∈ C −h x i be an < X -positive element. Then y is written as y = x m x j P ( X , H ) or y = x m h l P ( X , H ) where m ≥ l > j > P ( X , H ) is an {X , H} -positive word.Since x ∈ C , we may choose y so that m = 0 by considering x − m y instead.First we consider the case X 6⊂ h x i . Then we may choose y so that 1 In this section we briefly mention the computationalissue concerning the isolated ordering < X . Let G = hS | Ri be a group presentationand < G be a left ordering of G . The order-decision problem for < G is the algorithmicproblem of deciding for an element g ∈ G given as a word on S ∪ S − , whether ONSTRUCTION OF ISOLATED LEFT ORDERINGS 17 < G g holds or not. Clearly, the order-decision problem is harder than the wordproblem, since 1 < G g implies 1 = g . It is interesting to find an example of a leftordering < G of a group G , such that the order-decision problem for < G is unsolvablebut the word problem for G is solvable.There is another algorithmic problem which is also related to the order-decisionproblem of isolated orderings. We say a word on G ∪ G − is G -definite if w is G -positive or G -negative, or empty. If G defines an isolated ordering of G , then every g ∈ G admits a G -definite word expression. The G -definite search problem is aproblem to find a G -definite word expression of a given element of G . Theorem 2.18. Let us take G, H, X, < G , < H , z G , z H , G , H , X as in Theorem 1.1. (1) The order-decision problem for < X is solvable if and only if the order-decision problems for < G and < H are solvable. (2) The {X , H} -definite search problem is solvable if and only if the G -definitesearch problem and the H -search problem are solvable.Proof. Since the restriction of < X to G and H yields the ordering < G and < H respectively, if the order-decision problem for < X is solvable, then so is for < G and < H . Similarly, if {X , H} -definite search problem is solvable, then we are ableto get G -positive (resp. H -positive) word by transforming {X , H} -positive wordrepresenting elements of G (resp. H ) by using g i = x i ∆ H so G -definite ( H -definite)search problem is also solvable.The proof of converse is implicit in the proof of Theorem 1.1 (i). Recall thatin the proof of Property C (Proposition 2.17), we have shown that for a reducedstandard factorization F ( x ) = rp q · · · p l q l , x > X r ≥ H x < X r H < 1. Moreover, the proof of Property C (Proposition 2.17) is constructive,hence we can algorithmically compute an {X , H} -negative word expression of x if r < H G -definite search problem and the H -search problem is solvable.Thus, to solve the order-decision problem or {X , H} -definite search problem,it is sufficient to compute a reduced standard factorization. We have establishedtwo different methods to compute a reduced standard factorization, in the proofof Lemma 2.12 and Lemma 2.14. Both proofs are constructive, hence we canalgorithmically compute a reduced standard expression. (cid:3) It is not difficult to analyze the computational complexity of order-decision prob-lem or the {X , H} -definite search problems based on the algorithm obtained fromthe proof of Proposition 2.17, Lemma 2.12 and Lemma 2.14. In particular, weobserve the following results. Proposition 2.19. Let us take G, H, X, < G , < H , z G , z H , G , H , X as in Theorem1.1. (1) If the order-decision problems for < G and < H are solvable in polynomialtime with respect to the length of the input of words, then the order-decisionproblem for < X is also solvable in polynomial time. (2) If the G -definite search problem and the H -definite search problem are solv-able in polynomial time, then the {X , H} -definite search problem is alsosolvable in polynomial time. (3) Moreover, if one can always find a G -definite and an H -definite word ex-pression whose length are polynomial with respect to the length of the inputword, then one can always find an {X , H} -definite word expression whoselength is polynomial with respect to the length of the input word. Examples In this section we give examples of isolated left orderings produced by Theorem1.1. All examples in this section are new, and have various properties which pre-viously known isolated orderings do not have. For the sake of simplicity, in thefollowing examples we only use the infinite cyclic group Z , the most fundamentalexample of group having isolated orderings, as the basic building blocks.Other group with isolated orderings, such as groups having only finitely manyleft-orderings, or the braid group B n with the Dubrovina-Dubrovin ordering < DD ,also can be used to construct new examples of isolated orderings.3.1. Group having many distinct isolated orderings. Let a , . . . , a m ( m > m infinite cyclic groups Z ( i ) = h x i i . G = G a ,...,a m = ∗ Z Z ( i ) = h x , . . . , x m | x a = x a = · · · = x a m m i Recall that an infinite cycle group Z have exactly two left orderings, the standardone and its opposite. Using the standard left ordering for each factor Z ( i ) , byTheorem 1.1 we are able to construct an isolated left ordering < G so that therestriction of < G to the i -th factor Z ( i ) is the standard left ordering.First we give a detailed exposition of < G for the case m = 2 and m = 3. Example 3.1. (i) First we begin with the case m = 2, which was already considered in [3],[8]: G a ,a = Z (1) ∗ Z Z (2) = h x , x | x a = x a i By Theorem 1.1, we get an isolated ordering < G defined by the gener-ating set { x x − a , x } .(ii) Next we consider the case m = 3. There are two different ways to express G as an amalgamated free products of Z .(a) First we regard G a ,a ,a = G a ,a ∗ Z Z (3) = ( Z (1) ∗ Z Z (2) ) ∗ Z Z (3) .By (1), G a ,a have an isolated ordering defined by { x x − a , x } . Byapplying Theorem 1.1 again, we get the isolated ordering < ( •• ) • definedby { x x − a x − a , x x − a , x } .(b) Next we regard G a ,a ,a = Z (1) ∗ G a ,a = Z (1) ∗ ( Z (2) ∗ Z Z (3) ). Byapplying Theorem 1.1, we get the isolated ordering < • ( •• ) defined by { x x − a x x − a = x x − a x − a , x x − a , x } .Thus two orderings < ( •• ) • and < • ( •• ) derived from different factoriza-tions are the same ordering.As Example 3.1 (ii) suggests, the isolated orderings constructed from Theorem1.1 are independent of the way of factorization as amalgamated free products, thatis, the way of putting parenthesis in the expression Z (1) ∗ Z Z (2) ∗ Z · · · ∗ Z Z ( m ) . Allfactorizations give the same isolated ordering < G defined by { s , . . . , s m } , where s i is given by s i = x i x − a i +1 i +1 · · · x − a m m . ONSTRUCTION OF ISOLATED LEFT ORDERINGS 19 This is checked by induction on m . Take a factorization of G as G = G ∗ Z G = G a ,...,a k ∗ Z G a k +1 ,...,a m . By induction, the isolated ordering < of G is independentof a choice of a factorization of G , and is defined by s ′ i = x i x − a i +1 i +1 · · · x − a k k ( i = 1 , . . . , k ) . Similarly, the isolated ordering < of G is independent of a choice of a factorizationof G , and is defined by s ′′ j = x j x − a j +1 j +1 · · · x − a m m ( j = k + 1 , . . . , m ) . Thus by Theorem 1.1, we get an isolated ordering < G of G defined by s i = ( s ′ i x − a k +1 k +1 x k +1 x − a j +2 k +2 · · · x − a m m ( i = 1 , . . . , k ) ,x i x − a i +1 i +1 · · · x − a m m ( i = k + 1 , . . . , m )= x i x − a i +1 i +1 · · · x − a m m . The group G is the simplest example of groups with isolated orderings con-structed by Theorem 1.1. Nevertheless the group G and its isolated ordering < G have various interesting properties which have not appeared in the previous exam-ples: (1): The isolated ordering < G of G is not derived from Dehornoy-like orderingsif G is not generated by two elements. As we mentioned earlier, the special kind of left-orderings called Dehornoy-likeorderings produces isolated orderings, and all previously known examples of genuineisolated orderings are derived from Dehornoy-like orderings.In [3] it is proved that an isolated ordering derived from Dehornoy-like orderingshas a lot of convex subgroups: if the isolated orderings < H of a group H is derivedfrom the Dehornoy-like orderings, then there are at least r ( < H ) − < H -convex nontrivial subgroups. On the other hand Theorem 1.1 (vi) shows the isolatedorderings < G has only one proper, < G -convex nontrivial subgroup.If G is not generated by two elements, then r ( < G ) > 2. This implies that theisolated ordering < G of G is not derived from a Dehornoy-like ordering. This pro-vides a counter example of somewhat optimistic conjecture: every genuine isolatedordering is derived from Dehornoy-like ordering. (Recall that all previously knownexamples of genuine isolated orderings are constructed by Dehornoy-like orderings.)We remark that it is known that the group G = G a ,...,a m is a two-generatorgroup if and only if a i and a j are not coprime for some i = j [6]. Therefore forexample, the isolated ordering of G , , in Example 3.1 (ii) is an isolated orderingwhich is not derived from a Dehornoy-like ordering. (2): The natural right G -action on LO ( G ) has at least m − distinct orbitsderived from isolated orderings. There is a natural, continuous right G -action on LO( G ), defined as follows: Fora left ordering < of G and g ∈ G , we define the left ordering < · g by h ( < · g ) h ′ if hg < h ′ g . This action sends an isolated ordering to an isolated ordering. Althoughthis action is natural and important, little is known about the quotient LO( G ) /G . Recall that G is written as the amalgamated free products of m infinite cyclicgroups Z ( i ) . As we have seen, the way of decomposition of G ( the way of puttingparenthesis) does not affect the obtained isolated ordering < G .On the other hand, for a permutation σ ∈ S m , G a ,...,a m = G a σ (1) ,...,a σ ( m ) . Byviewing G a ,...,a m = G a σ (1) ,...,a σ ( m ) and applying the construction above, we get anisolated ordering < σ whose minimal positive element is x σ (1) x − a σ (2) σ (2) · · · x − a σ ( m ) σ ( m ) = x − ( m − a x σ (1) x σ (2) · · · x σ ( m ) . Thus, for two permutations σ and τ , if x σ (1) x σ (2) · · · x σ ( m ) and x τ (1) x τ (2) · · · x τ ( m ) are not conjugate, then two isolated orderings < σ and < τ belong to distinct G -orbits. Hence, we are able to construct ( m − G -orbits of isolated order-ings.Recall that these orderings are constructed from the standard left orderings of Z ( i ) . By using the opposite of the standard left-ordering of Z instead, we get other( m − G -orbits of isolated orderings in a similar way. Thus we have atleast 2( m − G -orbits derived from isolated orderings. (3): The natural right Aut ( G ) -action on LO ( G ) has at least ( m − distinctorbits derived from isolated orderings if all a , . . . , a m are distinct. As in the group G itself, there is a natural right Aut( G )-action on LO( G ). Fora left ordering < of G and θ ∈ Aut( G ), we define the left ordering < · θ by h < · θ g if hθ < gθ . The right G -action on LO( G ) can be regarded as the restriction of thenatural Aut( G )-action to the subgroup Inn( G ).There is one symmetry which reduces the number of orbits: the involution de-fined by x i x − i ( i = 1 , . . . , m ). This amounts to taking the opposite ordering.If all a , . . . , a m are distinct, φ ( a i ) = a ± j for any φ ∈ Aut( G ). Hence by a similarargument as (2), by looking at the minimal positive elements, we show that thereare ( m − G )-orbit derived from isolated orderings.Thus, the properties (2) and (3) show that the group G has quite a lot of essen-tially different isolated orderings.3.2. Centerless group with isolated ordering. Next we consider the construc-tion of the case z H is non-central. First of all, let G m,n = h b, c | b m = c n i . By Exam-ple 3.1 (i), G m,n has an isolated left ordering < m,n which is defined by { bc − n , c } .Let us consider a non-central element bc = bc − n · b m . Then it satisfies theinequality b m < m,n bc < m,n b m . Since b m is < m,n -cofinal central element, thisshows that bc is also < m,n -positive cofinal. < m,n is a ( bc − n )-right invariant ordering by Lemma 2.1, and < m,n is also a b m -right invariant ordering since b m is central. Thus, < m,n is a ( bc )-right invariantordering.Thus, we can take the non-central element bc as an element z H in Theorem 1.1and we are able to apply the partially central cyclic amalgamation construction.Now we consider the group H = H p,q,m,n = Z ∗ Z G m,n = Z ∗ Z ( Z ∗ Z Z ) defined by h a, b, c | b m = c n , a p = ( bc ) q i . This group has an isolated left ordering < H , defined by { a ( bc ) − q , bc − n , c } . Letus put x = a ( bc ) − q , y = ( bc ) − n , and z = c . Then the group H p,q,m,n is presented ONSTRUCTION OF ISOLATED LEFT ORDERINGS 21 as H p,q,m,n = h x, y, z | ( yz n − ) m = z n , ( x ( yz n ) q − ) p = ( yz n ) q i by using the generator { x, y, z } .Clearly, H has trivial center. This gives a first example of centerless group havingisolated orderings. In fact, Theorem 1.1 will allow us to construct many examplesof centerless group having isolated orderings. References [1] P. Dehornoy, I.Dynnikov, D.Rolfsen and B.Wiest, Ordering Braids, Mathematical Surveys andMonographs , Amer. Math. Soc. 2008.[2] T. Dubrovina and T. Dubrovin, On braid groups, Sb. Math, (2001), 693–703.[3] T. Ito, Dehornoy-like left orderings and isolated left orderings, J. Algebra (2013), 42–58.[4] V. Kopytov and N. Medvedev, Right-ordered groups, Siberian School of Algebra and Logic,Consultants Bureau, 1996.[5] P. Linnell, The space of left orders of a group is either finite or uncountable, Bull. LondonMath. Soc. (2011), 200–202.[6] S. Meskin, A. Pietrowski and A. Steinberg, One-relator groups with center, J. Austral. Math.Soc (1973), 319–323[7] A. Navas, On the dynamics of (left) orderable groups, Ann. Inst. Fourier, (2010), 1685–1740.[8] A. Navas, A remarkable family of left-ordered groups: Central extensions of Hecke groups, J.Algebra, (2011), 31–42.[9] C. Rivas, Left-orderings on free products of groups, J. Algebra (2012), 318–329.[10] A. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. , (2004),519-526. Department of Mathematics, The University of British Columbia, 1984 MathematicsRoad Vancouver, B.C, Canada V6T 1Z2 E-mail address ::