Recognizing the real line 7 1 0 2 A. M. W. Glass and John S. Wilson n a January 26, 2017 J 5 2 ] R Abstract G . Let (Ω,6) be a totally ordered set. We prove that if Aut(Ω,6) h is transitive and satisfies the same first-order sentences as Aut(R,6) t a (in the language of groups) then Ω and R are isomorphic ordered sets. m This improvement of a theorem of Gurevich and Holland is obtained [ as a consequence of a study of centralizers associated with certain 1 transitive subgroups of Aut(Ω,6). v 5 3 1 Introduction 2 7 0 In 1981, Gurevich and Holland [4] proved the following result. . 1 0 Theorem 1.1. Suppose that (Ω,6) is a totally ordered set such that Aut(Ω,6) 7 acts transitively on pairs (α,β) with α < β. If Aut(Ω,6) and Aut(R,6) satisfy 1 : the same first-order sentences, then Ω is isomorphic to R as an ordered set. v i X Other results of a similar kind were obtained in [2]. Theorem 1.1 required r first-order sentences in the language of ℓ-groups, which is richer than the language a of groups, but a slight extension shows that in fact only sentences in the language of groups are needed (see [1, Theorem 2B*]): Corollary 1.1. Suppose that (Ω,6) is a totally ordered set such that Aut(Ω,6) acts transitively on pairs (α,β) with α < β. If Aut(Ω,6) and Aut(R,6) satisfy the same first-order sentences in the language of groups, then Ω is isomorphic to R as an ordered set. 2010 AMS Classification: 20B07,06F15, 03C60, 05C05. Keywords: Transitive group, o-primitive, convex congruence, o-block, covering convex congruence. 1 Here we establish the following improvement to Theorem 1.1. Theorem A. Suppose that (Ω,6) is a totally ordered set on which Aut(Ω,6) acts transitively, and that Aut(Ω,6) and Aut(R,6) satisfy the same first-order sentences in the language of groups. Then Ω is isomorphic to R as an ordered set. Transitivity is necessary in theabove result. Let Λbeany rigid totally ordered setwith at least two elements (for example afinitetotally ordered set withat least two elements), and let Ω = Λ×R, with the order defined by (λ ,r ) < (λ ,r ) if 1 1 2 2 r < r or if both r = r and λ < λ . It is easy to see that Aut(Λ×R,6) is 1 2 1 2 1 2 isomorphic to Aut(R,6). Similar arguments allow us to strengthen other known results. Theorem B. Suppose that (Ω,6) is a totally ordered set on which Aut(Ω,6) acts transitively. If Aut(Ω,6) and Aut(Q,6) satisfy the same first-order sentences in the language of groups then Ω is isomorphic as an ordered set to Q or R\Q. The corresponding result with the stronger hypothesis that Aut(Ω,6) acts o-2 transitively on Ω, i.e., transitively on pairs (α,β) with α< β, is a slight extension of a result of Gurevich and Holland [4] (cf. [1, Theorem 2C*]). Thebreakthroughinthisworkcomesfromemployingatechniquein[6]: weuse doublecentralizers of certain subsets of groups of order-preservingautomorphisms of totally ordered sets Ω to give first-order expressibility of certain convex subsets of Ω. The ideas have implications for a large family of subgroups of the groups Aut(Ω,6) (see [3]). 2 Preliminaries and a reduction We write X ⊂ Y for X ⊆ Y and X 6= Y. Our notation for conjugates and commutators is in accordance with our use of right actions: we write gf for f−1gf and [f,g] for f−1g−1fg. AutomorphismgroupsAut(Ω,6)oftotallyorderedsets(Ω,6)areclosedunder taking the pointwise maximum f ∨g and pointwise minimum f ∧g of elements f,g defined, respectively, by α(f ∨g) = max{αf,αg} and α(f ∧g) = min{αf,αg} for all α∈ Ω. An ℓ-permutation group (G,Ω) is a subgroup of Aut(Ω,6) closed under the binary operations ∨ and ∧. Transitive ℓ-permutation groups are of particular interest, and all groups studied in this paper will be assumed to be transitive. Let (G,Ω) be a transitive ℓ-permutation group. A G-congruence on the set Ω is an equivalence relation C on Ω such that (αg)C(βg) whenever αCβ and g ∈ G 2 (α,β ∈ Ω). A convex G-congruence C on Ω is a G-congruence with all C-classes convex; these classes are called o-blocks. We suppress the mention of G if it is clear from context. By transitivity, each o-block ∆ is a class of a unique convex congruence; its set of classes is {∆g | g ∈ G}. We denote this convex congruence by κ(∆). Proposition 2.1. ([1,Theorem3.A])The setof convexcongruences of atransitive ℓ-permutation group is totally ordered by inclusion. If C and D are convex congruences with C ⊂D and there is no convex congru- ence strictly between C and D, then we say that D covers C and C is covered by D. Let α,β ∈ Ω be distinct. Then both the union U(α,β) of all convex congruences C for which α, β lie in distinct o-blocks and the intersection V(α,β) of all con- vex congruences C for which α, β lie in the same o-block are convex congruences. Clearly, U(α,β) is covered by V(α,β). Let K = {V(α,β) |α,β ∈ Ω,α6= β}. Thus K is totally ordered by inclusion. It is called the spine of (G,Ω). For all α,β ∈ Ω we have β = αg for some g ∈ G by transitivity. Therefore K can also be described as follows: K = {V(α,αg) | α ∈Ω,g ∈ G,αg 6=α}. Write T for the set of o-blocks of elements of K. If ∆ ∈T, then κ(∆) ∈K and so κ restricts to a surjective map from T to K. For each C ∈ K, write π(C) for both the convex congruence covered by C and its set of o-blocks; the latter inherits a total order from Ω. If ∆ is a C-class, let π(∆) be the set of all π(C)-classes contained in ∆. We define the stabilizer st(∆) and rigid stabilizer rst(∆) of an o-block ∆ as follows: st(∆):= {g ∈ G| ∆g =∆} and rst(∆) := {g ∈ G| supp(g) ⊆ ∆}, where supp(g) := {α ∈ Ω | αg 6= α}. So st(∆) and rst(∆) are convex sublattice subgroups of G and rst(∆) ⊆ st(∆). Each g ∈ st(∆) induces an automorphism g of the ordered set π(∆) given by ∆ Γg = Γg for all Γ ∈ π(∆). ∆ Let G(∆) := {g | g ∈ st(∆)}. ∆ 3 Note that (G(∆),π(∆)) is transitive and o-primitive. Furthermore, if K ∈ K ′ ′ ′ and ∆,∆ are both K-classes, then (G(∆),π(∆)) and (G(∆),π(∆ )) are isomor- ′ phic, an isomorphism being induced by conjugation by any f ∈ G with ∆f = ∆ since (Γf)(f−1gf) = (Γg)f for all g ∈ rst(∆), Γ ∈ π(∆). It is customary to write (G ,Ω ) for any of these ℓ-permutation groups; they are independent of K K the o-block ∆ of K to within ℓ-permutation isomorphism. For each g ∈ G and each subset Λ of Ω that is a union of convex g-invariant subsets of Ω, write dep(g,Λ) for the element of Aut(Ω,6) that agrees with g on Λ and with the identity elsewhere. We say that (G,Ω) is fully depressible if dep(g,Λ) ∈ G for all g ∈ G and all such Λ ⊆ Ω. In particular, if (G,Ω) is fully depressible, ∆ ∈ T and g ∈ st(∆), then dep(g,Λ) ∈ rst(∆) ⊆ G. Moreover, the action of {g | g ∈ rst(∆)} on π(∆) is equal to (G(∆),π(∆)) in this case for every ∆ ∆ ∈ T. Clearly Aut(Ω,6) itself is fully depressible. If G is transitive on all n-tuples (α ,...,α ) ∈ Ωn with α < ··· < α , we say 1 n 1 n that (G,Ω) is o-n transitive. We shall need the following result (see [1, Lemma 1.10.1]): Lemma 2.2. Every o-2 transitive ℓ-permutation group (G,Ω) is o-n transitive for all integers n > 2. We also need an immediate consequence of McCleary’s Trichotomy [5]: Proposition 2.3. Let (G,Ω) be a transitive fully depressible ℓ-permutation group. Then (G,Ω) is o-primitive if and only if either (I) (Ω,6) is order-isomorphic to a subgroup of the reals and the action of G on Ω is the right regular representation; or (II) (G,Ω) is o-2 transitive. Transitive o-primitive ℓ-permutation groups of type (II) are non-abelian. For each h∈ G, let Xh := {[h−1,hg] |g ∈ G} and Wh = [{Xhg | g ∈ G, [Xh,Xhg]6= 1}. ThesetsX ,W areevidently definableinthefirst-orderlanguageofgrouptheory. h h ForanysubsetS ofG,wewriteC2(S)asshorthandforC (C (S)),thedouble G G G centralizer of S in G. If S is definable in G in the first-order language of group theory then so is C2(S). G Proposition 2.4. Let (G,Ω) be a transitive fully depressible ℓ-permutation group. Then (G,Ω) is o-primitive if and only if C2(W ) = G for all g ∈ G\{1}. G g 4 This result has the following immediate consequence. Corollary 2.5. If (G ,Ω ), (G ,Ω ) are transitive fully depressible l-groups that 1 1 2 2 satisfy the same first-order sentences in the language of group theory, and one of these groups is o-primitive, then so is the other. We can now deduce Theorems A and B. Let Λ= R or Λ= Q. Then Aut(Λ,6) acts o-2-transitively on Λ and so it is o-primitive and non-abelian. Thus if (G,Ω) is a transitive fully depressible ℓ-permutation group satisfying the same first-order sentences (in the language of groups) as Aut(Λ,6), then (G,Ω) is non-abelian and G acts o-primitively on Ω by Proposition 2.4. Hence G acts o-2-transitively by Proposition 2.3. Theorems A and B now follow directly from Corollary 1.1 and the result [1, Theorem 2C*] cited in the Introduction. It remains now to prove Proposition 2.4. This will be an easy consequence of results in Sections 4 and 5. 3 A technical lemma Lemma 3.1. Let (G,Ω) be o-2 transitive and g,h ∈ G with supp(h)∩supp(hg)= ∅ and h 6= 1. Then there are elements f,k ∈ G such that [h−1,hf][h−g,hgk] 6= [h−g,hgk][h−1,hf]. Proof. Since supp(h) ∩supp(hg) = ∅, after interchanging h and hg if necessary, we may assume that there are supporting intervals ∆ ,∆ := ∆ g of h and hg, 1 2 1 respectively, such that δ < δ for all δ ∈ ∆ (i= 1,2). Without loss of generality, 1 2 i i δ h> δ for all δ ∈ ∆ (and so δ hg > δ for all δ ∈ ∆ ). Let γ,δ,λ,µ ∈ ∆ with 1 1 1 1 2 2 2 2 2 γ < γhg < µh−g < δ < λ <µ < δhg < λhg. The six elements γ, γhg, µh−g, δ, µ, λhg (1) constitute astrictly increasingsequencein∆1. Chooseξ−1,ξ0 ∈ ∆1 withξ−1 < ξ0, and elements ξ ,ξ ∈ ∆ with 1 2 2 ξ < ξ < ξ hg < ξ < ξ hg. 0 1 1 2 2 Then the six elements ξ−1, ξ0, ξ1, ξ1hg, ξ2, ξ2hg (2) constitute a strictly increasing sequence in Ω. Using o-6-transitivity we can find an element k of G that maps the nth element of sequence (2) to the nth element 5 of sequence (1) for each n. Since supp(h)∩supp(hg) = ∅ and ξ−1 ∈ ∆1 ⊆ supp(h) we have γhgk = γk−1hgk = ξ−1hgk = ξ−1k = γ. This and other similar easy calculations show that γhgk = γ, (γhg)hgk = γhg, (µh−g)hgk = δ, µhgk = λhg. Now choose α ∈ ∆ ⊆ supp(h) and β ∈ (αh−1,α), and choose ζ ,...,ζ ∈ 1 1 4 supp(h) such that the eight elements ζ , ζ , ζ , ζ , ζ h, ζ h, ζ h, ζ h 4 3 2 1 4 3 2 1 form a strictly increasing sequence. Since supp(h) < supp(hg), the eight elements βh−3, αh−3, βh−2, αh−2, γh−g, γ, δ, λ also form a strictly increasing sequence, and we can find an element f ∈ G that maps the nth term of the former of these two sequences to the nth term of the latter for each n. A routine calculation now shows that αh−2hf = λ, βh−2hf = δ, αh−3hf =γ, and βh−3hf = γh−g. Let w := [h−1,hf], and w := [h−g,hgk]. 1 2 Further simple calculations show that λw =γ and λw = δ. 1 2 Moreover, γw = γ and δw = βh−3hf = γh−g. 2 1 Hence λw w = γ 6= γh−g = δw = λw w . 1 2 1 2 1 4 Centralizers: the non-minimal case Throughout this section and the next, we assume that (G,Ω) is a fully depressible transitive ℓ-permutation group, and write T, K for its root system and spine. For each h∈ G, define X , W as in Section 2. For each ∆ ∈ T, let h h Q = {h ∈ rst(∆) | (∃α∈ Ω)(V(αh,α) = κ(∆)}. ∆ As (G,Ω) is transitive and fully depressible, we have Q 6= ∅. Since (rst(∆))g = ∆ rst(∆g) commutes with rst(∆) for g ∈/ st(∆), we also have X ⊆ rst(∆) and W ⊆ rst(∆) for all ∆ ∈ T and h∈ Q . h h ∆ We will use the following observation. 6 Remark 4.1. Let (Λ,6) be a totally ordered set and S be a finite set of pairwise disjoint convex subsets of Λ. If f ∈ Aut(Λ,6) and Sf = S, then Sf = S for all S ∈ S. For the rest of this section we assume that K has no minimal element. Lemma 4.2. Let ∆ ∈T and h ∈ Q . ∆ ′ ′ ′ ′ ′ (a) Let ∆ ∈ T with ∆ ⊂ ∆ and ∆h6= ∆, and let g ∈ rst(∆) with g 6= 1. (i) Then [h−1,hg] 6= 1. In particular, X 6=1. h (ii) If f ∈ G and [[h−1,hg],f] = 1, then ∆′f = ∆′. In particular, if ′ ′ f ∈ C (X ) then ∆f = ∆. G h (b) If β ∈ supp(h) and f ∈ G, then either βf = β or [[h−1,hg],f] 6=1 for some g ∈rst(∆). Proof. (a) The elements gh−1,g,gh have disjoint supports contained in ∆′h−1,∆′ and ∆′h respectively, and so the restrictions of [h−1,hg] = g−h−1gg−hg to these three sets are conjugates of g−1 and g2 and are non-trivial. Assertion (i) fol- lows. An arbitrary conjugate [h−1,hg]f has non-trivial restrictions to the dis- tinct o-blocks ∆′h−1f,∆′f and ∆′hf, and so if the hypothesis of (ii) holds then {∆′h−1,∆′,∆′h}f = {∆′h−1,∆′,∆′h}. Thus f must map each of ∆′h−1,∆′,∆′h to itself, by Remark 4.1. (b) Suppose that βf 6= β. By Proposition 2.1, one of the convex congruences ′ V(β,βh),V(β,βf) contains the other. Let ∆ ∈T be a non-singleton o-block that is strictly contained in the o-block containing β for each of these congruences. ′ ′ ′ ′ Then ∆ ⊂ ∆ and ∆f 6= ∆. Let g ∈ Q∆′; then g ∈ rst(∆) and from (a)(ii) we have [[h−1,hg],f] 6= 1. Lemma 4.3. Let ∆ ∈T and h ∈ Q . ∆ (a) C (X ) contains the pointwise stabilizer of ∆ and is contained in the point- G h wise stabilizer of supp(h). (b) Wh = [{Xhg | g ∈ st(∆)}. (c) C (W ) is the pointwise stabilizer of ∆. G h Proof. (a) The first inequality holds since X moves only points in ∆. h f Letf ∈ C (X ). SinceX ⊆ rst(∆)wehaveX = X ⊆ rst(∆)∩rst(∆f),and G h h h h since {∆g | g ∈ G} partitions Ω and X 6= 1 we have ∆f = ∆. Thus f ∈ st(∆). h Let β ∈ supp(h). Then βf = β by Lemma 4.2(b) since f ∈ C (X ). G h 7 (b) Let g ∈ G. If ∆g 6= ∆, then rst(∆)∩rst(∆g) = 1 and the elements of X h and Xhg have disjoint support. Thus [Xh,Xhg]= 1. Hence Wh = [{Xhg | g ∈st(∆),[Xh,Xhg]6= 1}. Now let g ∈ st(∆) and k = hg. So k ∈ Q . ∆ First suppose that there is some Γ ∈ π(∆) with Γh 6= Γ and Γk 6= Γ. Choose ′ ′ β ∈ Γ and ∆ ∈ T with β ∈ ∆ ⊂ Γ. We claim that there is an element y ∈ Q∆′ ′ ′ ′ ′ with βy 6= β. Choose β ∈∆ such that β, β belong to different π(∆)-classes. By ′ ′ ′ ′ ′ ′ transitivity there is an element y ∈ G with βy = β . Evidently ∆y = ∆, and ′ ′ the element y := dep(y ,∆) has the required properties. ′′ ′′ ′ ′′ ′′ Choose ∆ ∈ T with β ∈ ∆ ⊂ ∆ and ∆ 6= ∆ y. Arguing as above we can find x ∈ Q∆′′ with β ∈ supp(x). Write a := [k−1,kx] =x−k−1xx−kx. Thus supp(a) ⊂ (∆′′)k−1∪∆′′∪∆′′k ⊂ (∆′)k−1 ∪∆′∪∆′k, ′ and the unions above are disjoint unions since ∆ ⊂ Γ and Γk 6= Γ. On the classes ∆′′k−1,∆′′,∆′′k the element a agrees with x−k−1,x2,x−k respectively, none of which is the identity since x 6= 1. Therefore δ′a = δ′x2 for all δ′ ∈ ∆′. Since [h−1,hy] = y−h−1yy−hy we also have supp([h−1,hy]) ⊂ ∆′h−1∪∆′∪∆′h, and δ′[h−1,hy] = δ′y2 for all δ′ ∈ ∆′. Since supp(x)⊆ ∆′′ which is disjoint from ∆′′y2, for any δ′′ ∈ supp(x) we have δ′′[h−1,hy]a = δ′′y2x2 = δ′′y2 6= δ′′x2y2 = δ′′a[h−1,hy]. But [h−1,hy] ∈ X and a ∈ X . Hence [X ,X ]6= 1 and X ⊆ W . h k h k k h Now suppose instead that Γk = Γ or Γh = Γ for all Γ ∈ π(∆). Then (G(∆),π(∆)) cannot be abelian and so is of type (II) in Lemma 2.3. Lemma 3.1 gives elements of X and X whose images in (G ,π(∆)) fail to commute, and h k ∆ again we conclude that X ⊆ W . k h (c) The pointwise stabilizer of ∆ lies in C (W ) since W ⊆ rst(∆). G h h Let δ ∈ ∆ and α ∈ supp(h). Choose g ∈rst(∆) with αg = δ. So δ ∈ supp(hg). By (a), CG(Xhg) fixes each point of supp(hg) and so fixes δ. Since CG(Wh) ⊆ CG(Xhg) by (b), we conclude that CG(Wh) fixes δ. The assertion follows. Proposition 4.4. Let ∆ ∈ T. Then C2(W ) = rst(∆) for each h ∈ Q . In G h ∆ particular, C2(W ) is independent of the choice of h∈ Q : G h ∆ C2G(Wh)= C2G(Wh′) for all h,h′ ∈ Q∆. 8 Proof. By Lemma 4.3(c) the subgroups rst(∆) and C (W ) commute and so G h rst(∆)⊆ C2(W ). We must prove that C2(W ) ⊆rst(∆). G h G h Suppose that z ∈ G with ∆z 6= ∆. So ∆z ∩ ∆ = ∅. Since (G,Ω) is fully depressible, rst(∆) contains an element x 6= 1; then xz ∈ rst(∆z) and so [x,z] 6= 1. Thus z 6∈C2(W ) by Lemma 4.3(c). It follows that C2(W ) ⊆ st(∆). G h G h Hence if g ∈ C2(W ), then the element g := dep(g,∆) of rst(∆) is defined. G h 0 From above, rst(∆) ⊆ C2(W ) and so f := gg−1 ∈ C2(W ). Suppose that f 6= 1. G h 0 G h ′ ′ Let α ∈ supp(f) and ∆ ∈ T be the o-block of V(α,αf) with α ∈ ∆. Let ′′ ′ ′′ ′′ ′′ ′′ ∆ ⊂ ∆ with α ∈ ∆ and ∆ f 6= ∆ , and let y ∈ Q∆′′. Since supp(y) ⊆ ∆ but supp(yf) ⊆ ∆′′f we have [y,f] 6= 1. However y ∈ C (W ) and f ∈ C2(W ), and G h G h we have a contradiction. Hence f = 1 and g = g ∈ rst(∆). 0 5 Centralizers: the minimal case Again let (G,Ω) be a fully depressible transitive ℓ-permutation group with spine K. For thecasewhenthespineKof(G,Ω)hasaminimalelement weneedanextra condition. We say that a transitive ℓ-permutation group (G,Ω) is locally abelian if its spine K has a minimal element K and the o-primitive ℓ-permutation group 0 (G(∆),π(∆)) is abelian for each o-block ∆ of K . 0 For the rest of this section we assume that the spine K of G has a minimal element and that (G,Ω) is not locally abelian. The results in the previous section can all be recovered under the above hy- potheses on (G,Ω). This follows from the following observation, in which we write f 6 g for f ∨g =g and H for {g ∈G | g >1}: + Remark 5.1. Let(H,Λ)bean o-2transitiveℓ-permutation group,andµ ,µ ∈ Λ 1 2 with µ < µ . Let γ ,γ ∈ Λ with µ < γ < γ < µ . By o-3 transitivity, there is 1 2 1 2 1 1 2 2 f ∈ H withγ f = γ andµ f = µ for i= 1,2. Soµ = µ fn < γ fn < µ fn = µ 1 2 i i 1 1 1 2 2 for all n ∈ Z. Let Ξ be the smallest convex subset of Λ containing {γ fn | n∈ Z}. 1 Then Ξf = Ξ. Let g := dep(f,Ξ). Then g ∈ Aut(Λ,6) with supp(g) ⊆ (µ ,µ ) + 1 2 and γ g = γ . 1 2 The next three results extend the corresponding results (Lemmata 4.2 and 4.3 and Proposition 4.4) of Section 4. Lemma 5.2. Let ∆ ∈ T and h ∈ Q . Let g ∈ rst(∆) and suppose that there ∆ + is α ∈ supp(h) such that supp(g) ⊆ (α,αh) if α < αh or supp(g) ⊆ (αh,α) if αh < α. (a) (i) [h−1,hg] 6= 1; in particular X 6= 1. h 9 (ii) If f ∈ G and [[h−1,hg],f] = 1, then supp(ghi)f = supp(ghi) for i = 0,±1. (b) If f ∈ G, then either βf = β for all β ∈ supp(h) or [[h−1,hg],f] 6= 1 for some g ∈ rst(∆) . + Proof. (a) We assume that α < αh, the proof when αh < α being similar. Write c := [h−1,hg] = g−h−1gg−hg. For i = 0,±1 we have supp(ghi) ⊆ (αhi,αhi+1). These intervals are pairwise disjoint and supp(c) lies in their union. Since supp(c) may not be convex and the intervals may not be mapped to them- selves by f, we cannot apply Remark 4.1. This is where we use that g > 1. The restriction of c to (α,αh) is g2 > 1 and c is strictly positive only on supp(g). Moreover, if [c,f] = 1, then (c ∨ 1)f = cf ∨ 1f = c ∨ 1, so f must conjugate g2 = c ∨ 1 ∈ G to itself, and c−1 ∨ 1 to itself. Since f is order-preserving and supp(gh−1) < supp(g) < supp(gh) and supp(g2) = supp(g), we must have supp(ghi)f = supp(ghi) for i = ±1. (b) Suppose that βf 6= β for some β ∈ supp(h) and that [h−1,hg] and f commute for all g ∈ rst(∆) . Let K be the minimal element of K and Λ be the + 0 K o-block with β ∈ Λ ⊆ ∆. Then (G(Λ),π(Λ)) is o-primitive and o-2 transitive 0 since(G,Ω)is notlocally abelian. Now βf andβharedistinctfromβ andsothere is an interval (β ,β ) containing β and disjoint from (β f,β f) and (β h,β h)∪ 1 2 1 2 1 2 (β h−1,β h−1). By Remark 5.1 with µ = β and µ = β , there is g ∈ G 1 2 1 1 2 2 + with β ∈ supp(g) ⊆ (β ,β ). Thus the sets supp(ghi) for i ∈ {0,±1} are pairwise 1 2 disjointandsince[h−1,hg] = g−h−1gg−hgweobtainthatβ[h−1,hg]f = βg2f > βf. Since [h−1,hg], f commute, it follows that βf[h−1,hg]> βf. Thus βf ∈ supp([h−1,hg]) ⊆ supp(g)∪supp(gh)∪supp(gh−1). Since supp(g) ⊆ (β ,β ) and βf ∈ (β f,β f) we have βf ∈/ supp(g), whereas if βf ∈ supp(gh) 1 2 1 2 then βf[h−1,hg] = βfg−h < βf, and if βf ∈ supp(gh−1), then βf[h−1,hg] = βfg−(h−1) < βf. A contradiction ensues and the lemma is proved. Lemma 5.3. Let ∆ ∈T and h ∈ Q . ∆ (a) C (X ) contains the pointwise stabilizer of ∆ and is contained in the point- G h wise stabilizer of supp(h). (b) Wh = [{Xhg | g ∈ st(∆)}. (c) C (W ) is the pointwise stabilizer of ∆. G h 10