Non-discrete Euclidean Buildings for the Ree and Suzuki groups

NON-DISCRETE EUCLIDEAN BUILDINGS F OR THE REE AND SUZUKI GR OUPS PETRA HITZELBERG ER, LINUS KRAMER AND RICHARD M. WEISS Abstract. W e call a non-discrete Euclidean building a Bruh at-Tits sp ac e if its automorphism gr oup contains a subgroup that induces the subgroup generated b y all the r oot groups of a ro ot datum of the buil ding at inﬁnity . This is the class of non-discrete Euclidean buildings int ro duced and studied by Bruhat an d Tits in [2]. W e give the complete classiﬁcation of Bruhat-Tits spaces whose building at inﬁnity is the ﬁxed p oint s et of a p olarity of an ambien t building of t yp e B 2 , F 4 or G 2 associated with a Ree or Suzuki group endow ed with the usual ro ot datum. (In the B 2 and G 2 cases, this ﬁxed point set is a building of rank one; in the F 4 case, it is a generalized octagon whose W e yl group i s not crystallographic.) W e al so show that each of the se Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at inﬁnit y is the corresp onding ambien t building. 1. Introduction Suppo se that X is an irreducible aﬃne building. Typically (and f or certain if the dimensio n o f X is at least three), the automo r phism gro up of the building a t inﬁnit y of X contains subgroups constituting a r o ot datum deﬁned over a ﬁeld K . In this c a se the aﬃne building X is uniquely determined by a v aluatio n of this r o ot datum whic h is, in tur n, uniquely determined by a dis crete v alua tio n of K . The notion of a v aluation o f the ro ot datum o f a spherical building makes p er- fectly goo d sense if w e drop the requirement that the v alues lie in a discrete subgroup of R . In the non-discr ete case, there is no lo nger a co rresp onding aﬃne building X . There do es exist, how ever, an analogo us structure called a non- discr ete Euclide an building . Non-discrete Euclidean buildings w ere ﬁrst in tro duced and studied b y Bruhat and Tits in [2]. These structure s w ere ﬁrst axio matized and, in dimension greater than tw o , classiﬁed b y Tits in [12]. (Other fundamental referenc e s ab o ut non- discrete Euclidean buildings are [4] and [6]; see also [1] and [8].) Non-discrete Euclidea n buildings a re sometimes calle d aﬃne R -buildings (since in the simples t cas e they ar e R -trees) or ap artment systems (since they have a part- men ts but are not really buildin gs) or, as in [4], simply Euclide an buildings , although this term is more commonly synonymous with “aﬃne building.” In 4.13 below, we propose the term Bruhat-Tits sp ac e for the class of non- dis crete Euclidean buildings that were int ro duced and studied b y Bruhat and Tits in [2]. (The term Bruhat-Tits sp ac e was used in [5] to descr ibe c omplete metric spaces that A ddr ess of the ﬁrst t wo authors: F ach b ereich Mathematik und Informatik, Universit¨ at M ¨ unster, Einsteinstrasse 62, 48149 M ¨ unster, Germany . A ddr ess of the thir d author: Departmen t of Mathematics, T ufts Unive rsi t y , 503 Boston Ave nue, Medford, M A 02155, USA. 1 2 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS satisfy a ce r tain “semi-par allelogr am rule” intro duced by S. Lang. This is a mo re general class o f metric s pa ces which includes not o nly all non-discrete Euclidea n buildings but also, for example, all simply connected Riemannia n manifolds o f non- po sitive curv a tur e. W e mention to o that Bruhat-Tits spaces in o ur sense are not necessarily complete as metric spaces; see Section 7.5 in [2].) Suppo se now that ∆ is the spherical building (o f r ank one or tw o) asso ciated with a Ree or Suzuki group. T hus ∆ is the ﬁxed p oint set of a “p olar it y” o f a building of t yp e B 2 , F 4 or G 2 which is deﬁned o ver a ﬁeld extension K/F , where p := char( K ) equals 2 in the ﬁrst tw o cases and 3 in the third case, and the po larity is deﬁned in terms o f a Tits endomorphism θ of K (as deﬁned in 5.1) whose image is F . On page 173 of [12], Tits remark s that an a rbitrary v aluation ν of K (where “ arbitrar y” means “arbitra ry non-trivia l real-v alued non- archimedean”) extends to a v alua tion of the ro ot datum of ∆—and thus there exists a corresp o nding Bruha t-Tits space whose building at inﬁnity is ∆—if and only if the v aluation ν is θ -inv ariant. It is the go al of this pap er to make this statement mo re pr ecise and to ﬁll in all the details justifying it. Here θ -invariant means that ν ◦ θ is e quivalent to ν , i.e. that ν ( x ) ≥ 0 for x ∈ K ∗ if and only if ν ( x θ ) ≥ 0. This is the same as saying that ν ( x θ ) = γ · ν ( x ) for some po sitive real num b er γ a nd all x ∈ K ∗ . Since θ is a Tits e ndo morphism of K , it follows that ν is θ -inv a riant if and only if ν ( x θ ) = √ p · ν ( x ) for all x ∈ K ∗ . Thus, in pa r ticular, ν canno t b e θ - in v a riant if it is discrete (since the ratio of t wo v alues of a discrete v a luation is alwa ys r ational). 2. O ver view Contin uing to put pre cision aside for a mo men t, we can summarize the main results of this paper roughly as follows; see also 8 .11. Theorem 2.1. L et G b e a Re e or Suzuki gr oup. Then the fol lowing hold: (i) Ther e exists a Moufang bu ilding ˙ ∆ of typ e B 2 , F 4 or G 2 having a p olarity ρ deﬁne d ove r a p air ( K , θ ) as describ e d in Se ction 1 such that G is the gr oup induc e d by the c ent r alizer C ˙ G ( ρ ) on the set ˙ ∆ ρ of ﬁx e d p oints of ρ , wher e ˙ G is the su b gr oup of Aut( ˙ ∆) gener ate d by the r o ot gr oups of ˙ ∆ . (ii) The set ∆ := ˙ ∆ ρ has the structu r e of a Moufang building of r ank one in c ases B 2 and G 2 , of r ank two whose Weyl gr oup is dihe dr al of or der 16 in c ase F 4 . (iii) F or e ach valuation ν of the ﬁeld K , ther e exists a unique n on- discr ete Eu- clide an buildi ng ( ˙ X , ˙ A ) determine d by ν whose build ing at inﬁnity is ˙ ∆ and whose automorphism gr oup induc es ˙ G on ˙ ∆ . (iv) L et ν and ( ˙ X , ˙ A ) b e as in (iii). Th en ther e exists an automorphism ˙ ρ of ( ˙ X , ˙ A ) inducing the p olarity ρ on ˙ ∆ if and only if ν is θ -invariant. F urt hermor e, ˙ ρ , if it exists, is unique. (v) If ν , ( ˙ X , ˙ A ) and ˙ ρ ar e as in (iv), then ther e is a c anonic al non-discr ete Euclide an building ( X , A ) c ontaine d in the ﬁxe d p oint set of ˙ ρ in ( ˙ X , ˙ A ) whose building at inﬁnity is ∆ and whose automorphism gr oup c ontains a sub gr oup inducing G on ∆ . NON-DISCRETE EUCLIDEAN BUILDINGS 3 (vi) Every n on-discr ete Euclide an building ( X , A ) whose building at inﬁn ity is ∆ and whose automorphism gr oup c ont ains a sub gr oup inducing G on ∆ arises fr om a θ -invariant valuation ν of K as describ e d in (v). Pr o of. Assertion (i) is es s ent ially the deﬁnition of a Ree o r Suzuki gro up. The building ˙ ∆ and its p olarity ρ are describ ed in 5 .3. Ass ertion (ii) is prov ed in 6 .5.iii and 6 .7. Ass ertions (iii) and (iv) follow from 4.16 a nd 5.1 5. Ass e r tion (v) is proved in 7.1 1 a nd as s ertion (vi) is a conseq ue nce of 7.1. ✷ This pap er is org a nized as follows: In Sections 3– 4 we rev iew the basic r e sults ab out ro ot data, v a luations o f ro ot data and no n-discrete Euclidean buildings we require. In Sections 5 we int ro duce the spherica l buildings o f type B 2 , F 4 and G 2 having p olarities that give rise to the Ree and Suzuki g r oups and in Section 6 we a ssemble the basic pr o p e rties of the subbuildings ﬁxed by these p ola r ities we require. Our main results—7.1, 7.4 and 7 .1 1—are then proved in Sections 7 and 8. Throughout this pap er w e will use ∆ and simila r letters to denote spherica l buildings a nd ( X, A ) and simila r letters to denote Euclidean buildings. 3. R oot da t a and v a lua tio ns W e now start paying attention to the details. In this section w e review the notions of a root datum o f a spherical building and a v a lua tion o f a ro ot datum. Notation 3.1 . Let ( W , S ) b e an irr educible spherical Coxeter sys tem. Then either W can be identiﬁed with the W eyl group o f a n ir reducible ro ot system Φ so that S consists of the reﬂectio ns determined b y the elements in a basis, or | S | = 2 and W is a dihedral group of order 2 n for n = 5 or n > 6. In the latter cas e we let Φ consist of 2 n vectors evenly distributed around the unit circle in a 2-dimensiona l Euclidean s pa ce and think of S a s the tw o reﬂections determined by t wo of these vectors making an a ngle of ( n − 1)18 0 /n deg rees. (Later we will r efer to this set Φ as I 2 ( n ).) In bo th cases, we denote by A the am bient Euc lidea n space of Φ and by Aut(Φ) the g roup of isometrie s o f A mapping Φ to itself. Note that ( W, S ) is uniquely deter mined by Φ. When we sometimes call W the Weyl gr oup of Φ (as is usual), we re a lly have the pair ( W , S ) in mind. Notation 3. 2. Let ( W, S ) and Φ b e as in 3.1. F or all α, β ∈ Φ such that α 6 = ± β , the interval ( α, β ) is the s - tuple ( γ 1 , γ 2 , . . . , γ s ) of all elements γ i ∈ Φ such that for some positive real n um b ers p i and q i (whic h depend o n α and β ), (3.3) γ i / | γ i | = p i α/ | α | + q i β / | β | , where ∠ ( α, γ i ) < ∠ ( α, γ j ) if a nd only if i < j . Note that s dep ends on the pa ir α and β and tha t for some pair s, s = 0. T o deﬁne the interv a l ( α, β ), we could have omitted the denominato r s in 3.3. W e included the denominators b ecause the coeﬃcients p i and q i in this equation (as it is wr itten) a re needed in the statement o f condition (V2) in 3.14. Notation 3.4. Let ( W , S ), Φ and A b e as in 3.1. T o each α ∈ Φ we a sso ciate the reﬂection s α given by s α ( v ) = v − 2( v · α ) α for each v ∈ A . A wall of Φ is the ﬁxed p oint s et of one o f these reﬂection. A Weyl chamb er o f Φ is a connected comp onent of A with all the walls removed. W e call 4 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS the clo sure o f a W eyl cham b er a se ctor o f Φ. The sector s ar e p olyhedra l cones; a fac e o f Φ is a face of one o f these cones. The s et of a ll faces o f Φ forms a s implicial complex called the Coxeter c omplex of ( W, S ) (or Φ). W e will denote this simplicial complex b y Σ(Φ). T o eac h e lement α of Φ we ass o ciate the half-space H α := { v ∈ A | v · α ≥ 0 } . A r o ot o f the Coxeter complex is the set of faces co ntained in one of these ha lf- spaces. The map α 7→ H α th us g ives rise to a cano nic a l bijection from Φ to the set of r o ots o f Σ(Φ). Con v ent ions 3. 5. Let ∆ be an irreducible spherical building. Then every apa rt- men t of ∆ is iso morphic to the Coxeter complex Σ(Φ) for some Φ as in 3.1. The corres p o nding Coxeter sys tem ( W , S ) is us ua lly called the t yp e o f ∆. W e pr efer, instead, to say tha t ∆ is of typ e Φ. (Th us a building of type B ℓ is the same thing as building of t yp e C ℓ .) Let Σ be an a partment of ∆. A r o ot of Σ is the image of a r o ot of Σ(Φ) under an iso mo rphism from Σ(Φ) to Σ. Thus for each such isomo r phism we hav e a ca nonical bijection fro m Φ to the set of ro ots of Σ. W e will usually ass ume that an isomo r phism from Σ(Φ) to Σ is ﬁxed and identify Φ with the set of ro ots of Σ via this bijection. Deﬁnition 3. 6. Let ∆ b e an ir reducible spherical building of rank at least tw o. F or each ro ot α of ∆ (i.e. o f some a partment of ∆), let U α be the intersection of the stabilizers in Aut(∆) of a ll the chambers that are co nt ained in so me pa nel o f ∆ which con tains tw o cham b er s in α . It follows fro m Coro llary 3.14 in [1 4] that U α acts trivially on the set of cham b ers in α itself. (The subgroups of the form U α are called the r o ot gr oups of ∆.) The building ∆ is Moufang (equiv alently , “∆ satisﬁes the Mo ufang c ondition ”) if the following hold: (i) ∆ is thick (i.e. every panel con tains at least three chambers). (ii) F or each ro o t α of ∆, the ro ot group U α acts transitively on the set of apartments containing α . Tits showed that thick irreducible spherical buildings o f ra nk at least thre e , as well as all the ir reducible residues of rank tw o of such a building, always satisfy the Moufang condition; see, for example, Theorems 11.6 and 1 1.8 in [14] for a pr o of. Irreducible spher ic al buildings of rank a t least tw o satis fying the Moufang co ndition were clas siﬁed in [10] and [13]. See [1 5, 30.14] for a summar y of the r esults. Prop ositio n 3.7. L et ∆ b e an irr e ducible spheric al bu ilding of r ank ℓ ≥ 2 satisfying the Moufang pr op erty. Then for e ach r o ot α of ∆ , the r o ot gr oup U α acts sharply tr ansitively on the set of ap artments c ontaining α . Pr o of. This is prov ed, for example, in Theo rem 9.3 and Prop os ition 11.4 of [14]. ✷ Now suppo se tha t ∆ is a building of rank one. In other words, ∆ is simply a set (whose elements are the cham b ers of ∆) and Aut(∆) is the full s y mmetric gro up on ∆. The apartments of ∆ ar e the tw o-e le men t subsets of ∆ and thus the r o ots of ∆ are just the o ne-element s ubsets o f ∆. Normally w e will use letters like x and y to na me element s of ∆; when w e w a nt to emphasize that an elemen t of ∆ is b eing considered a s a ro ot, ho wev er, we will give it a name like α or β . Deﬁnition 3.8 . Let ∆ b e a building of rank o ne, i.e. of type Φ := A 1 , let Σ be an apartment of ∆ and let the tw o elements of Σ b e identiﬁed with the tw o elemen ts NON-DISCRETE EUCLIDEAN BUILDINGS 5 of Φ. A Mo ufang structu r e o n ∆ is a collection ( U α ) α ∈ Φ of non- trivial subgroups of Aut(∆) such tha t the followin g hold: (i) F or each α ∈ Φ, the subgroup U α ﬁxes α a nd acts sharply transitively on ∆ \{ α } ; and (ii) F or e a ch α ∈ Φ, the subgro up U α is norma liz e d by the s tabilizer of α in the group G := h U α , U − α i . The gr oups U g α for a ll α ∈ Φ and all g ∈ G ar e calle d r o ot gr oups . A Moufang structure on ∆ is indep endent of the choice of Σ a nd its ident iﬁcation with Φ up to conjugation in the gro up G . Since the ro ot gr o ups are required to b e no n-trivial, ∆ can have a Mo ufang structure only if it is thick, i.e. if | ∆ | ≥ 3. When we say that ∆ is Moufang , we mean that we hav e a par ticula r Moufang structur e on ∆ in mind (whose ro ot groups we will alwa ys call U α , U β , etc.). Note that with this conv ention, 3 .7 holds also when ℓ = 1 (with “sa tis fying the Moufang prop erty” int erpr e ted as meaning “ having a Moufang structure with ro ot groups U α ). Remark 3.9. Let ∆ b e a n irreducible spher ic al building of ra nk ℓ ≥ 2 which satisﬁes the Moufang condition and let P be a pa ne l of ∆ viewed as a set of cham b er s . F or each cham b er x in P , let α b e an arbitr ary ro ot of ∆ containing x but no other cham b er in P . By 3.7, U α acts faithfully on P . F urthermore, the per mut ation gr oup induced by the r o ot group U α on P is indep endent of the choice of α . (This follows fro m P rop osition 11.11 in [14].) Hence every r ank o ne residue of ∆ inher its a canonical Moufang structure fro m ∆ whose ro ot g roups ar e isomo rphic to r o ot g roups of ∆. Prop ositio n 3. 10. L et ∆ b e an irr e ducible spheric al building of typ e Φ satisfying the Moufang c ondition as deﬁne d in 3.6 and 3.8 and let Σ b e an ap artment of ∆ (to which we apply 3.5). Then for e ach α ∈ Φ (i.e. to e ach r o ot α of Σ ), the fol lowing hold: (i) Ther e exist maps λ and κ fr om U ∗ α to U ∗ − α such t hat for e ach u ∈ U ∗ α , the pr o du ct m Σ ( u ) := κ ( u ) uλ ( u ) maps Σ to itself and induc es t he unique r eﬂe ction s α deﬁne d in 3.4 on Φ (which inter changes the r o ots α and − α ). (ii) m Σ ( u ) − 1 = m Σ ( u − 1 ) for e ach u ∈ U α . (iii) m Σ ( κ ( u )) = m Σ ( λ ( u )) = m Σ ( u ) for al l u ∈ U α . Pr o of. The ﬁr st asser tion is a consequence of 3.7 and the o ther tw o fo llow fro m the ﬁrst; s ee, for example, in [13, 6.1–6.3 ]. ✷ Prop ositio n 3.1 1. L et ∆ b e an irr e ducible spheric al building satisfying the Mo- ufang c ondition as deﬁne d in 3.6 and 3.8 and let G † b e the sub gr oup of Aut(∆) gener ate d by al l the r o ot gr oups of ∆ . Then G † acts tr ansitively on the set of al l p airs (Σ , C ) , wher e Σ is an ap artment of ∆ and C is a chamb er of Σ . Pr o of. This is proved, for example, in Prop os ition 11.12 of [14]. ✷ Note in the rank one case, 3.11 just says that G † is a 2-tra nsitive p er m utation gr oup on ∆. 6 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS Deﬁnition 3. 12. Let ∆ b e an irreducible spherical building of type Φ satisfying the Moufang c ondition and let Σ b e a n a pa rtment of ∆ (to which we apply 3.5). F or ea ch α ∈ Φ, let U α be the corr esp onding ro ot gro up. The r o ot datum of ∆ (based a t Σ) is the pair (Σ , ( U α ) α ∈ Φ ). Let ∆ b e as in 3.12. By 3.11, the ro ot datum of ∆ is, up to conjugation in the group G † , indep endent of the c hoice of the apartment Σ. No te that a ro ot datum and a Mo ufa ng str ucture o n ∆ (as deﬁned in 3.8) ar e es s ent ially the same thing when ∆ has rank one. By [13, 40.17], ∆ is uniquely determined by its ro o t datum when the rank of ∆ is a t leas t t wo. Theorem 3. 1 3. L et ∆ b e an irr e ducible spheric al building of typ e Φ satisfying the Moufang c ondition, let t he n otion of t he int erval fr om one element of Φ to another b e as in 3.2 and let Σ b e an ap artm ent of ∆ (to which we apply 3.5). Then for al l or der e d p airs α, β of elements of Φ su ch that β 6 = ± α , [ U α , U β ] ⊂ U γ 1 U γ 2 · · · U γ s , wher e ( γ 1 , γ 2 , . . . , γ s ) is the interval ( α, β ) , if t he interval ( α, β ) is not empty and [ U α , U β ] = 1 if it is. Pr o of. This is proved in [7, 6.1 2(ii)]. ✷ Deﬁnition 3. 14. Let ∆ b e an irreducible spherical building of type Φ satisfying the Moufang c ondition, let Σ b e an apartment of ∆ (to which we apply 3 .5) and let (Σ , ( U α ) α ∈ Φ ) be the ro ot datum of ∆ based a t Σ as deﬁned in 3.12. A valuation of this ro ot datum is a collectio n ϕ := ( ϕ α ) α ∈ Φ of no n-constant maps ϕ α from U ∗ α to R such that the following hold: V1: F or each α ∈ Φ and ea ch k ∈ R , the set U α,k := { u ∈ U α | ϕ α ( u ) ≥ k } is a s ubgroup of U α , where we assign ϕ α (1) the v alue ∞ (so that 1 ∈ U α,k for all k ). V2: F or all α, β ∈ Φ such that α 6 = ± β and for all k , l ∈ R , [ U α,k , U β ,l ] = U γ 1 ,p 1 k + q 1 l U γ 2 ,p 2 k + q 2 l · · · U γ s ,p s k + q s l , where γ i , p i , q i and s are as in 3.3. V3: F or all α, β ∈ Φ, all u ∈ U ∗ α and all g ∈ U β , the quantit y t := ϕ s α ( β ) ( g m Σ ( u ) ) − ϕ β ( g ) is independent g and if α = β , then t = − 2 ϕ α ( u ). Here s α is as in 3.4 and m Σ ( u ) is as in 3.1 0.i. Note that the condition (V2) is v ac uo us when the rank of ∆ is one. Deﬁnition 3.15. Let ∆, Φ and Σ b e as in 3.14 and supp os e that ϕ and ψ are tw o v aluations of the ro ot datum of ∆ based a t Σ. Then ϕ a nd ψ are e quip ol lent if for some x in the ambien t Euclidean space A of the ro o t sys tem Φ, ϕ α ( u ) = ψ α ( u ) + α · x for all α ∈ Φ and all u ∈ U ∗ α (in whic h case w e write ϕ = ψ + x ). NON-DISCRETE EUCLIDEAN BUILDINGS 7 Prop ositio n 3 .16. L et ∆ , Φ and Σ b e as in 3.14 and su pp ose that ϕ and ψ ar e two valuations of the r o ot datum of ∆ b ase d at Σ such t hat ϕ α = ψ α for s ome α ∈ Φ . Then ψ and ϕ ar e e quip ol lent (as deﬁne d in 3.15). Pr o of. This holds by Prop osition 6 in [1 2]. F or more details , see Theor em 3.41 in [15]. ✷ Deﬁnition 3. 17. Let ∆, Φ and Σ b e as in 3.14, let ϕ = ( ϕ α ) α ∈ Φ be a v aluatio n of the r o ot da tum of ∆ ba sed at Σ and let ρ b e a n automor phis m of ∆ mapping Σ to itself. W e will say that ϕ is ρ -invariant if ϕ α ( u ) = ϕ α ρ ( u ρ ) for all α ∈ Φ and all u ∈ U ∗ α . Prop ositio n 3.18. L et ∆ , Φ and Σ b e as in 3.14, let ϕ b e a valuation of t he r o ot datum of ∆ b ase d at Σ , let w b e an element of the r o ot gr oup U ∗ α such that ϕ α ( w ) = 0 and let m 0 = m Σ ( w ) . Then for e ach α ∈ Φ , ϕ α ( g m 0 m Σ ( u ) ) − ϕ α ( g ) = 2 ϕ α ( u ) for al l g , u ∈ U ∗ α . Pr o of. Let β b e the ro ot opp osite α in Σ. Le t g , u ∈ U ∗ α and let v = κ ( u ) ∈ U ∗ β , where κ is a s in 3.1 0.i. Then m Σ ( u ) = m Σ ( v ) by P rop osition 11.2 4 in [14], ϕ β ( v ) = − ϕ α ( u ) b y P rop osition 3 .25 in [15] (or [7, 10.10]) a nd ϕ α ( y m Σ ( v ) ) = ϕ β ( y ) − 2 ϕ β ( v ) for all y ∈ U β by condition (V3) (with both ro ots equal to β ). Thus ϕ α ( g m 0 m Σ ( u ) ) = ϕ α ( g m 0 m Σ ( v ) ) = ϕ β ( g m 0 ) − 2 ϕ β ( v ) = ϕ β ( g m 0 ) + 2 ϕ α ( u ) By a nother applica tion of condition (V3) (this time with b oth ro ots equal to α ) and the choice of w , we have ϕ β ( g m 0 ) = ϕ α ( g ) . ✷ 4. Non-discrete Euclidean Buildings In this s ection we ass emb le a few basic facts ab out non-dis c rete Euclidean build- ings. W e start with the deﬁnition. Notation 4.1. Let W , A and Φ b e a s in 3.1. Let W denote the gr oup generated by W and the g roup T consisting of all trans la tions of A . Thus W is a gr oup of isometries o f A . Mo reov er, W = T W and T is a normal subgr o up of W . Deﬁnition 4.2. Let Φ and ( A , W ) be a s in 4.1 a nd let H α for all α ∈ Φ b e as in 3.4. Let X b e a set and let A b e a family of injections of A into X . The elements of A will b e ca lle d cha rts and the images of c harts will b e called ap art m ents . Sets of the form f ( H α ) for some c hart f and some α ∈ Φ will b e called r o ots of ( X , A ) and sets of the form f ( S ) for s ome c hart f and so me se ctor (resp ectively , face) of Φ (as deﬁned in 3.4) will b e called se ctors (resp ectively , fac es ) of ( X, A ). The pa ir 8 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS ( X, A ) is a non-discr ete Euclide an building of typ e Φ if the fo llowing six axioms hold: A1 : If f ∈ A a nd w ∈ W , then f ◦ w ∈ A . A2 : If f , f ′ ∈ A , then the set M := { v ∈ A | f ( v ) ∈ f ′ ( A ) } is clo sed a nd conv ex and there exists w ∈ W such that the ma ps f and f ′ ◦ w coincide on M . A3 : Every tw o points o f X a re con tained in a common apartment. A4 : If S and S ′ are sectors of ( X , A ), then there exists an apartment containing sectors S 1 and S ′ 1 such that S 1 ⊂ S and S ′ 1 ⊂ S ′ . A5 : Three apar tmen ts which int erse c t pair wise in ro ots hav e a non- empt y in- tersection. A6 : There is a metr ic d on X such that for all v, z ∈ A and a ll f ∈ A , d ( f ( v ) , f ( z )) equals the Euclidean distance betw e e n v and z . If ( X , A ) is a no n-discrete Euclidean building of type Φ, we call the pair ( A , W ), which is uniquely determined by Φ, its mo del and we deﬁne the dimension o f ( X , A ) to b e the dimension o f A . By (A3), the metric d in (A6) is unique. V arious equiv alent deﬁnitions of a non- discrete Euclidean building (and a pr o of of their equiv alence) ca n b e found in Theore m 1.2 1 of [6]. See also P rop osition 2.2 1 of [6]. Deﬁnition 4 . 3. Let ( X, A ) and ( X ′ , A ′ ) be tw o non-discrete Euclidean buildings having the same t y p e Φ. An isomorphism ψ from ( X, A ) to ( X ′ , A ′ ) is a bijection from X to X ′ such that A ′ = { ψ ◦ f | f ∈ A} . W e denote by Aut( X , A ) the g roup of all isomorphisms fr om ( X , A ) to itself. Deﬁnition 4.4. Let ( X , A ) b e a non- dis crete Euclidea n building of type Φ, let ( A , W ) b e the model of ( X , A ), let τ b e an isometry of A norma lizing the group W and let A τ = { f ◦ τ | f ∈ A} . Then ( X , A τ ) is a non-dis c r ete Euclidean building of t yp e Φ with the same under- lying metric structur e as ( X , A ) . Now suppose in addition that τ 2 = 1. Then an isomorphism fro m ( X , A ) to ( X , A τ ) is automatically an isomo rphism from ( X, A τ ) to ( X , A ). W e call such an is o morphism a τ -automorphism of ( X, A ). Deﬁnition 4. 5. W e say that tw o non-discrete E uclidean buildings ( X , A ) and ( X ′ , A ′ ) ar e e qu ivalent (o r one is a dilation of the other) if they hav e the same type Φ and therefore the same model ( A , W ) , X = X ′ and A ′ = { f ◦ δ | f ∈ A} for some dilation δ o f A (where dilation means multiplication by a non-zer o con- stant). Thus equiv alent no n-discrete E uclidean buildings hav e the s ame underlying metric s tr ucture up to a co nstant p ositive facto r . Deﬁnition 4.6. Let ( X , A ) b e a no n-discrete Euclidean building with mo del ( A , W ) and let x, x ′ ∈ X . By (A3), ther e exists a chart f and po int s x 1 , x ′ 1 ∈ A such that f ( x 1 ) = x and f ( x ′ 1 ) = x ′ . The int erval [ x, x ′ ] is the image under f of the in terv a l [ x 1 , x ′ 1 ]. By (A2), the in terv a l [ x, x ′ ] is independent o f the c hart f . By the CA T(0) NON-DISCRETE EUCLIDEAN BUILDINGS 9 prop erty (prov ed, for example, in Prop osition 2.10 of [5]), the in terv a l [ x, x ′ ] is, in fact, the unique geo desic co nnecting x to x ′ . Notation 4.7. Let ( X , A ) be a non- discrete Euclidea n building of type Φ. Tw o faces F a nd F ′ of ( X, A ) (as deﬁned in 4.2) are called p ar al lel if they are at ﬁnite Hausdorﬀ distance, i.e. if b oth sup x ′ ∈ F ′ d ( x ′ , F ) and sup x ∈ F d ( x, F ′ ) are ﬁnite. B y (A6), this is an equiv alence relation on faces. F or each face F o f ( X, A ), w e denote b y F ∞ the corresp onding parallel class and fo r each apartment A of ( X, A ), w e denote b y A ∞ the set of par allel classes of faces containing a face of A . If b and b ′ are tw o parallel c la sses, we set b ≤ b ′ whenever sup x ′ ∈ F ′ d ( x ′ , F ) < ∞ . for a ll F ∈ b and all F ′ ∈ b ′ . This notion makes the set of para llel cla sses of faces into a s implicial complex. W e deno te this simplicial c o mplex by ( X , A ) ∞ . By Prop ositio n 1 in [12] (see also Prop er ty 1.7 in [6]), ( X , A ) ∞ is, in fact, a spherical building of t yp e Φ (a s deﬁned in 3.4) and the map A 7→ A ∞ is a bijection from the set of apar tmen ts of ( X , A ) to the set of apar tment s o f ( X , A ) ∞ . The building ( X, A ) ∞ is called the building at inﬁ n ity of ( X, A ). It is irr educible (since it is of t yp e Φ) and its rank is the same as the dimensio n of ( X, A ). Notation 4.8. Let ( X , A ) be a non-discre te Euclidean building o f type Φ with mo del ( A , W ), let A b e an apa r tment of ( X, A ), let x b e a p oint of A and let Σ = A ∞ . Let Σ(Φ) and H α (for each α ∈ Φ) b e as in 3 .4 and suppo se that Φ is ident iﬁed with the set of ro o ts of Σ via an isomorphism from Σ(Φ) to Σ as descr ib e d in 3.5 . Then there e xists a unique c hart f s uch that the follo wing hold: (i) f ( A ) = A . (ii) f (0) = x . (iii) F or ea ch α ∈ Φ, a secto r S of Φ is c ontained in the ha lf-space H α if and only if the cham b er f ( S ) ∞ of Σ is cont ained in the ro ot α of Σ. Let f A,x denote the chart f . Remark 4.9. Let Φ a nd ( A , W ) be as in 4.1. Then ( A , W ) is a non-discrete Euclidean building of type Φ with just one a partment. It thu s has a building at inﬁnit y whose faces are of the form F ∞ for some face F of Φ. Notation 4.10. Let ( X , A ) b e a non- discrete Euclidean building of type Φ and let ( A , W ) b e its mo del. Let F b e a face of ( X , A ). Then F is the ima g e o f a face of Φ (as deﬁned in 3.4) under some c hart f . The vertex of F is the imag e of the origin of A under f ; b y (A2), this notion is indep endent o f the c hoice of f . Now let x be a p oint of X . W e declare t wo p oints y and z of X \{ x } to be e quivalent at x if [ x, y ] ∩ B = [ x, z ] ∩ B for some op en ba ll B centered at x , where [ x, y ] and [ x, z ] are int erv als as deﬁned in 4.6. This is a n equiv alence r elation o n X \{ x } . F or each y ∈ X \{ x } , let g x ( y ) denote its equiv alence class. W e declar e tw o fac e s F and F 1 with vertex x to b e 10 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS equiv alent if g x ( F ) = g x ( F 1 ). By (A2), this holds if and only if F ∩ B = F 1 ∩ B for some ope n ball B centered at x . The equiv a lence cla ss of a face F is called the germ of F and a germ at x is the g e rm of a face with vertex x . The set o f germs at x form a simplicial complex o n the set g x ( X \{ x } ). As observed in Section 1.3 of [6], this simplicial complex is a building o f type Φ whose apartments are the sets g x ( A ) for all apartments A of ( X, A ) containing x . W e call this building the r esidue of (∆ , A ) at x and denote it by (∆ , A ) x . (The residue at x is called the bu ilding of dir e ct ions at x in [6].) The residues of a non-discrete E uclidean building ( X , A ) are no t necess arily thic k, a nd the r e s idues at diﬀerent po int s of X are not necessar ily is omorphic to each other . They are all, how ever, of t y pe Φ. Notation 4.11. Let ( X , A ) be a non-dis c rete E uc lide a n building, let x ∈ X , let the residue ( X , A ) x of ( X , A ) a t x b e as in 4.10 a nd let f b e an element o f A mapping the origin 0 to x . Then there exis ts a unique type-pr eserving iso morphism from the Coxeter complex Σ(Φ) (as deﬁned in 3.4) to an apartment of the residue ( X , A ) x which maps ea ch face F of Φ to the ge rm a t x containing f ( F ). W e denote this isomorphism by f ∗ . F or the r est of this section, w e exa mine the spe c ia l case that the building at inﬁnit y of a non-discrete Euclidean building is Moufang. Theorem 4.12. L et ( X, A ) b e a non-discr ete Euclide an building of typ e Φ with mo del ( A , W ) such that the building at inﬁ nity ∆ := ( X , A ) ∞ satisﬁes the Moufang pr op erty as deﬁne d in 3.6, let ℓ denote the dimension of ( X, A ) , let A b e an ap art- ment of ( X , A ) , let Σ = A ∞ , let x b e a p oint of A , let Σ(Φ) b e as in 3.4 and let H α,k = { v ∈ A | v · α ≥ k } for al l α ∈ Φ and al l k ∈ R . L et Φ b e identiﬁe d with the set of r o ots of Σ via an isomorphi sm fr om Σ(Φ) to Σ as describ e d in 3.5 and let f := f A,x b e as in 4.8. If ℓ ≥ 2 , then the fol lowing hold: (i) F or every r o ot α ∈ Φ , ther e exists a c anonic al inje ction fr om the r o ot gr oup U α into Aut( X , A ) such t hat for e ach u ∈ U α , its image under t his inje ction induc es u on ∆ . (ii) F or every α ∈ Φ , ther e exists a map ϕ α fr om the r o ot gr oup U ∗ α of ∆ to R such that Fix A ( u ) = A ∩ A u = f ( H α,ϕ α ( u ) ) for e ach u ∈ U ∗ α . Pr o of. See Sectio ns 10 and 11 of [12]. ✷ Note that under the hypotheses of 4.1 2, w e alwa ys iden tify each roo t group U α with its image under the injection in 4.12.i. Th us, in particular , the u in 4.12.ii is really the canonical image in Aut( X , A ) of an element u ∈ U α . With the following deﬁnition we describ e those non- discrete Euclidean building s which were s tudied in [2]. Deﬁnition 4.13. A Bruhat-Tits sp ac e is a no n-discrete E uclidean building ( X , A ) such that the follo wing ho ld: (i) The spher ical building ∆ := ( X , A ) ∞ is Mo ufa ng (in the sense of 3.6 or 3.8). NON-DISCRETE EUCLIDEAN BUILDINGS 11 (ii) The conclusions of 4.12 ho ld. Let ( X, A ) be a non-discrete Euclidean building sa tisfying (i) and let ℓ denote the dimension of ( X , A ). If ℓ = 1 , saying that ∆ is Moufang means we have a particular Moufang structure on ∆ in mind, a nd (ii) is to be interpreted with re s pe c t to this Moufang structure. If ℓ ≥ 2, then (ii) holds a utomatically . A non-disc r ete Euclidean building of r ank ℓ ≥ 3 alwa ys satisﬁes 4.13.i. (This is prov ed, for example, [13, 40.3 ].) Thus in dimension three o r higher , “non-dis crete Euclidean building” and “Bruhat-Tits space” are the s a me thing. Con v ent ion 4.14. Let ∆ b e a Moufang building o f rank one in the s e ns e of 3.8. When w e say that “∆ is the building at inﬁnit y of the Br uha t-Tits spa ce ( X , A ),” we mean that the conclusions of 4.12 hold with respect to the particular Moufang structure on ∆ w e ha ve in mind. The follo wing r esults o f Bruhat-Tits and Tits a re fundamen ta l. Theorem 4.15. L et ( X , A ) , A , x , ∆ , Σ and ϕ α for α ∈ Φ b e as in 4.12. Then ϕ := { ϕ α | α ∈ Φ } is a valuation of the r o ot datum of ∆ b ase d at Σ . Mor e over, the valuation ϕ is indep endent of t he choic e of the p oint x in A up to e quip ol lenc e (as deﬁne d in 3.15). Pr o of. This is the ﬁrst par t of Theorem 3 in [12]. ✷ Theorem 4. 1 6. L et ∆ b e an irr e ducible spheric al building of typ e Φ satisfying the Moufang c ondition, let Σ b e an ap artm en t of ∆ (to which 3.5 is applie d) and let ϕ b e a valuation of the r o ot datum of ∆ b ase d at Σ as deﬁne d in 3.14. Then ther e exists a Bruhat-Tits sp ac e ( X , A ) of typ e Φ , an ap artment A of ( X , A ) and a p oint x A of A such that the fol lowing hold: (i) ∆ is the building at inﬁnity of ( X, A ) ∞ (in the sense of 4.14 if the r ank of ∆ is one) and Σ = A ∞ . (ii) F or e ach α ∈ Φ , ϕ α is the map which app e ars in 4.12.ii when 4.12 is applie d to the triple ( X , A ) , A and x A . If ( X ′ , A ′ ) , A ′ and x ′ A is a se c ond tr iple with these pr op erties, then ther e exists an isomorphi sm fr om ( X , A ) to ( X ′ , A ′ ) mapping A to A ′ and x A to x ′ A . Pr o of. Existence is prov e d in Section 7.4 o f [2] and uniquenes s in Prop osition 6 o f [12]. ✷ 5. The Ree an d Su zuki gr oups In this section we collect a few well known facts ab out the Ree a nd Suzuki groups. All of these results ar e co nt ained mo re or less explicitly in [9] and [11]. There ar e three families o f Ree a nd Suzuki gro ups. Beg inning in 5.1 and for the rest of this paper , we will refer to three cases which we will call “case B ,” “case F ” and “ case G .” Notation 5.1. Let K b e a ﬁeld of characteristic p os itive p and supp ose that θ is a Tits endomorphism of K . This means that θ is a n endomo rphism of K such that θ 2 is the F rob enius map x 7→ x p . Thus, in par ticular, F := K θ is a subﬁeld of K iso morphic to K which contains the s ubﬁeld K p . Supp ose , to o, that p = 2 in 12 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS cases B and F and p = 3 in case G . In ca se B let L b e an a dditiv e subgr oup of K containing F such that L · F ⊂ L (so L is a vector s pa ce over F ) and K = h L i (where h L i denotes the subring of K gener ated by L ) and let Λ deno te the indiﬀerent s e t ( K, L, L θ ) as deﬁned in [1 3, 10.1 ]. In case F let Λ deno te the comp osition algebr a ( K, F ) as deﬁned in [15, 30.17]. In ca s e G let Λ denote the hexago nal system ( K/F ) ◦ as deﬁned in [13, 1 5.20]. Let ∆ denote the building B D 2 (Λ) in case B , the building F 4 (Λ) in ca se F and the building G 2 (Λ) in ca se G in the notation descr ib e d in [1 5, 30.17]. Th us B D 2 (Λ) is the Mo ufang quadrangle ca lled Q D (Λ) in [13, 16 .4] and G 2 (Λ) the Moufang hexa gon c a lled H (Λ) in [13, 16.8]. The t yp e Φ of the building ∆ is B 2 in case B , F 4 in case F a nd G 2 in ca se G . (Alterna tively , we can deﬁne ∆ to b e the unique building o f type Φ whose r o ot datum is as describ ed in 5.3 b elow.) The building ∆ in 5.1 is split if and only if K is p erfect; if K is not p e rfect, ∆ is simply a building o f mixed t yp e (as deﬁned, for example, in [15, 30.24]). The follo wing element τ plays a central role from now on. Notation 5.2 . Let Φ be as in 5.1, let A b e the a mbien t spa ce of Φ a nd let S b e a sector of Φ. There is a unique non-trivia l ele ment of Aut(Φ) (as deﬁned in 3.1) ﬁxing S . This automor phism induces a no n-type-preser ving automorphism o f the Coxeter co mplex Σ(Φ) and has or der tw o. W e denote it by τ . Theorem 5 .3. L et ∆ , Φ , K , L , etc. b e as in 5.1, let τ and S b e as in 5.2 , let Σ b e an ap artment of ∆ , let C b e a chamb er of Σ , let ψ b e the unique sp e cial isomorphism fr om Σ(Φ) to Σ mapping S to C and let Φ b e identiﬁe d with the set of r o ots of Σ via ψ as indic ate d in 3.5. Then for e ach α ∈ Φ , ther e exists an isomorphi sm x α fr om the additive gr oup of L in c ase B , r esp e ctively, the additive gr oup of K in c ases F and G , to the r o ot gr oup U α of ∆ su ch that the c ol le ction ( x α ) α ∈ Φ has the fol lowing pr op erties: (i) Ther e exists a unique automorphi sm ρ of ∆ mapping the p air ( C , Σ) to itself such t hat x α ( t ) ρ = x τ ( α ) ( t ) for al l α ∈ Φ and al l t ∈ K (or al l t ∈ L ). (ii) In c ases B and F , [ U α , U β ] = 1 whenever ∠ ( α, β ) ≤ 90 ◦ , [ x α ( s ) , x β ( t )] = x α + β ( st ) for al l s, t ∈ K whenever ∠ ( α, β ) = 1 2 0 ◦ and [ x α ( s ) , x β ( t )] = x √ 2 α + β ( s θ t ) x α + √ 2 β ( st θ ) for al l s, t ∈ K (or al l s, t ∈ L ) whenever ∠ ( α, β ) = 13 5 ◦ . (iii) In c ase G , ther e exists p ar ameters ǫ, ǫ 1 , . . . , ǫ 4 such that [ U α , U β ] = 1 whenever ∠ ( α, β ) ≤ 90 ◦ , [ x α ( s ) , x β ( t )] = x α + β ( ǫst ) NON-DISCRETE EUCLIDEAN BUILDINGS 13 for al l s, t ∈ K whenever ∠ ( α, β ) = 1 2 0 ◦ and [ x α ( s ) , x β ( t )] = x √ 3 α + β ( ǫ 1 s θ t ) x 2 α + √ 3 β ( ǫ 2 s 2 t θ ) · · x √ 3 α +2 β ( ǫ 3 s θ t 2 ) x α + √ 3 β ( ǫ 4 st θ ) for al l s, t ∈ K whenever ∠ ( α, β ) = 15 0 ◦ . The p ar ameters ǫ, ǫ 1 , . . . , ǫ 4 ar e al l e qual t o + 1 or − 1 ; their values dep en d on t he or der e d p air ( α, β ) bu t not on s or t ; and their values ar e as in 5.4–5.6 if α, β b oth c ontain the chamb er C . (iv) In al l t hr e e c ases, x β ( t ) m Σ ( x α (1)) = x s α ( β ) ( ± t ) for al l α, β ∈ Φ and al l t ∈ K (or L ). Pr o of. Suppo se ﬁr st that we are in cas e G and let the ro ots o f Φ b e n umbered α 1 , . . . , α 12 going around the origin clo ckwise, where the indices are to b e read mo dulo 12. The set of r o ots of Σ can be identiﬁed with Φ so that α i contains the cham b er C if and only if i ∈ [1 , 6]. Th us τ ( α i ) = α 7 − i for a ll i . By [1 3, 1 6.8], [ U α i , U α j ] = 1 whenev er i, j ∈ [1 , 6] and i < j ≤ i + 3 and ther e exis t isomorphisms x i for each i ∈ [1 , 6] from the additive g roup of K to the ro ot g r oup U α i such that for all s, t ∈ K , (5.4) [ x 1 ( s ) , x 6 ( t )] = x 2 ( − s θ t ) x 3 ( − s 2 t θ ) x 4 ( s θ t 2 ) x 5 ( st θ ) , as w e ll as (5.5) [ x 1 ( s ) , x 5 ( t )] = x 3 ( − st ) and (5.6) [ x 2 ( s ) , x 6 ( t )] = x 4 ( st ) . By [13, 7.5], ther e exists a unique automorphism ρ o f ∆ mapping ( C, Σ) to itself such that x i ( t ) ρ = x 7 − i ( t ) for all i ∈ [1 , 6 ] and for all t ∈ K . Let H be the cham b erwise stabilizer of Σ in Aut(∆) a nd let Q b e the subgr oup of H consisting of tho se element s g such that for eac h ro o t α o f Σ, either g centralizes U α or g inv er ts every element o f U α . Let m i = m Σ ( x i (1)) for i = 1 and 6 and let N = h m 1 , m 6 i . By [11, 2.9(6)] and [13, 29.12 ], | Q | = 4 and the kernel of the actio n o f N on Σ is N ∩ Q and N / N ∩ Q ∼ = D 12 . F or each i ∈ [7 , 1 2], there thus is a unique shor tes t word g in m 1 and m 6 mapping α j to α i , where j = 1 if i is o dd a nd j = 6 if i is even. W e set x i ( t ) = x j ( t ) g for all t ∈ K . By [1 3, 6.2], m ρ 1 = m 6 and m ρ 6 = m 1 . It follo ws that x i ( t ) ρ = x τ ( i ) ( t ) for all t ∈ K a nd for all i ∈ [7 , 12]. Thus (i) holds . Let α, β ∈ Φ. By [13, 6.4] (or [11, 2.9]), there exists g ∈ N mapping α to β such that x α ( t ) g = x β ( ± t ) for all t ∈ K ∗ . Since g is unique up to an elemen t of N ∩ Q , it follows that x α ( t ) g = x β ( ± t ) for all t ∈ K ∗ and for all g ∈ N mapping α to β . Thu s (iv) holds (in c ase G ). By 5.4 – 5.6, it follows that also (iii) holds. The proo f in case B is virtually the s a me, except that we cite [13, 16.4 ] in place of [13, 1 6.8] and use the observ ation that this time m Σ ( u ) 2 = 1 for all α ∈ Φ and all u ∈ U ∗ α by 3 .10.ii (since a ll the ro ot gro ups ar e of exponent tw o ). In ca se F , the 14 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS pro of in the previous tw o cas e s can b e suitably mo diﬁed (using [15, 3 2.14] in place of [13, 1 6.4]). W e do not give the details, howev er, since the cla ims in cas e F also follow from the observ ations in [11, 1.1–1.2]. ✷ Deﬁnition 5.7 . Let ∆ a nd ρ b e as in 5.3, let ∆ ρ denote the s et of cham b ers ﬁxed by ρ and let G † be as in 3.11. Let G b e the g roup induced on ∆ ρ by the centralizer of ρ in G † . The group G is commonly ca lled Suz( K, θ , L ) in case B (these are the Suzuki gr oups ), Ree( K , θ ) in case G (these a r e the Re e gr ou ps ) and 2 F 4 ( K, θ ) in case F . The groups 2 F 4 ( K, θ ) were also discov ered by Ree a nd are a lso sometimes called Ree gr oups, but these groups are now mor e commonly asso ciated with Tits due to his thorough study of them in [11]. The name Suz( K , θ , L ) is us ua lly abbre viated to Suz( K , θ ) w he n L = K . W e use the rest of this sectio n to prove a r esult (5 .15) ab out v aluations of the ro ot datum of the building ∆. Prop ositio n 5. 8. L et ∆ , Φ , Σ and ( x α ) α ∈ Φ b e as in 5.3, let α, β ∈ Φ , let t ∈ K , let u ∈ K ∗ (or t ∈ L and u ∈ L ∗ in c ase B ) and let h ( u ) = m Σ ( x α (1)) m Σ ( x α ( u )) . Then the fol lowing hold: (i) In c ases B and F , x β ( t ) h ( u ) = x β ( u − θ t ) if ∠ ( α, β ) = 45 ◦ and x β ( t ) h ( u ) = x β ( u θ t ) if ∠ ( α, β ) = 135 ◦ . (ii) In c ase G , x β ( t ) h ( u ) = x β ( u θ t ) if ∠ ( α, β ) = 30 ◦ and x β ( t ) h ( u ) = x β ( u − θ t ) if ∠ ( α, β ) = 150 ◦ . (iii) In al l t hr e e c ases, x β ( t ) h ( u ) = x β ( u − 2 t ) if α = β , x β ( t ) h ( u ) = x β ( u − 1 t ) if ∠ ( α, β ) = 60 ◦ , x β ( t ) h ( u ) = x β ( t ) if ∠ ( α, β ) = 90 ◦ , x β ( t ) h ( u ) = x β ( ut ) if ∠ ( α, β ) = 120 ◦ and x β ( t ) h ( u ) = x β ( u 2 t ) if α = − β . NON-DISCRETE EUCLIDEAN BUILDINGS 15 Pr o of. By 3.10.i, h ( u ) acts trivially on Σ and hence normalizes U β . The claims hold for β 6 = ± α b y parts (ii) and (iii) of 5.3 and [11, 2.9]. Suppo se that we can c ho ose β ∈ Φ suc h that ∠ ( α, β ) = 120 ◦ . By 5.3, we have (5.9) [ x α ( t ) , x β ( s )] = x α + β ( ǫst ) and (5.10) [ x α + β ( s ) , x − α ( t )] = x β ( ǫ ′ st ) for all s ∈ K , where ǫ, ǫ ′ = ± 1. W e know that x β (1)) h ( u ) = x β ( u ) and x α + β ( t ) h ( u ) = x α + β ( u − 1 t ) . Setting s = 1 in 5.9 and conjuga ting by h ( u ), we o btain [ x α ( t ) h ( u ) , x β ( u )] = x α + β ( ǫu − 1 t ) . Comparing this identit y with 5.9 itself, we co nclude that x α ( t ) h ( u ) = x α ( u − 2 t ). Set- ting s = 1 in 5.10 a nd co njugating by h ( u ), we conclude similar ly that x − α ( t ) h ( u ) = x − α ( u 2 t ). Suppo se next that there is no β ∈ Φ such that ∠ ( α, β ) = 12 0 ◦ . Then w e are in case B , so c ha r( K ) = 2, and we ca n ﬁnd β such that ∠ ( α, β ) = 135 ◦ . Thus (5.11) [ x α ( t ) , x β ( s )] = x √ 2 α + β ( t θ s ) x α + √ 2 β ( ts θ ) and (5.12) [ x √ 2 α + β ( s ) , x − α ( t )] = x α + √ 2 β ( s θ t ) x β ( st θ ) by 5 .3.ii. W e know tha t h ( u ) centralizes U 3 , x β (1) h ( u ) = x β ( t θ ) and x √ 2 α + β ( t θ ) h ( u ) = x √ 2 α + β ( u − θ t θ ) . Setting s = 1 in 5.11 and co njugating by h ( u ), we o bta in [ x α ( t ) h ( u ) , x β ( u θ )] = x √ 2 α + β ( u − θ t θ ) x α + √ 2 β ( t ) . Comparing this identit y with 5.11 itself, we co nclude ag ain tha t x α ( t ) h ( u ) = x α ( u − 2 t ) . Setting s = 1 in the identit y 5.12 and co njugating by h ( u ), we conclude similarly that x − α ( t ) h ( u ) = x − α ( u 2 t ) also in this cas e. ✷ Corollary 5.13. L et ∆ , Φ , Σ and ( x α ) α ∈ Φ b e as in 5.3, let G † b e as in 3.11 and let H b e the chamb erwise stabilizer of Σ in G † . Then for e ach α ∈ Φ and for e ach h ∈ H , ther e exists z ∈ K ∗ (or z ∈ F ∗ in c ase B , wher e F = K θ ) such t hat x α ( t ) h = x α ( z t ) for al l t ∈ K (or al l t ∈ L ). Pr o of. By [13, 33.9 ], H = h m Σ ( x α (1)) m Σ ( x α ( u )) | α ∈ Φ and u ∈ K ∗ (or L ∗ ) i . The cla im holds, therefore, by 5.8. ✷ Notation 5.14. Let Φ b e as in 5.1. Let Φ = Φ 0 ∪ Φ 1 be the pa rtition of Φ into t wo subsets Φ 0 and Φ 1 such that tw o elements of Φ ar e in the sa me subset if and only if they ha ve the same length. 16 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS Prop ositio n 5.15. L et ∆ , Φ , Σ , ρ and ( x α ) α ∈ Φ b e as in 5.3, let Φ 0 and Φ 1 b e as in 5.14, let ν b e a (r e al value d) valuation of the ﬁeld K and let p = char( K ) . L et ϕ α ( x α ( t )) = ν ( t ) for al l α ∈ Φ 0 and al l t ∈ K ∗ (or al l t ∈ L ∗ ), let ϕ α ( x α ( t )) = ν ( t θ ) / √ p for al l α ∈ Φ 1 and al l t ∈ K ∗ (or al l t ∈ L ∗ ) and let ϕ = ( ϕ α ) α ∈ Φ . Then the fol lowing hold: (i) ϕ is a valuation of the r o ot datum of ∆ b ase d at Σ . (ii) ϕ is ρ -invariant (as deﬁne d in 3.17) if and only if ν is θ -invariant. Pr o of. The collectio n ϕ sa tisﬁes condition (V1) in 3 .14 by the deﬁnition of a v alua- tion. By parts (ii) and (iii) of 5.3 and some calcula tion, ϕ satisﬁes co ndition (V2). Cho ose α, β ∈ Φ, u ∈ U ∗ α and g ∈ U ∗ β . Supp ose ﬁrst that α = β . By 5.3.iv, ϕ − α ( g m Σ ( u ) ) = ϕ α ( g m Σ ( u ) m Σ ( x α (1)) − 1 ) . By 3 .10.ii, therefo r e ϕ − α ( g m Σ ( u ) ) = ϕ α ( g ( m Σ ( x α (1)) m Σ ( − u )) − 1 ) . Hence ϕ − α ( g m Σ ( u ) ) − ϕ α ( g ) = − 2 ϕ α ( u ) by 5 .8.iii. By 5.3.iv and 5.8 (a nd a similar calculation), the quantit y ϕ s α ( β ) ( g m Σ ( u ) ) − ϕ β ( g ) is indep endent of g whenever β 6 = α . T hus ϕ satisﬁes condition (V3). Hence (i) holds. By 5 .3.i, ϕ is ρ -in v ar iant if a nd only if ν ( t θ ) / √ p = ν ( t ) for all t ∈ K ∗ . Thus (ii) ho lds. ✷ 6. Spherical Buildings for the Ree and Suzuk i groups In this section we show tha t the G -set ∆ ρ deﬁned in 5.7 has the structure of a spherical building satisfying the Moufang condition whos e ro o t g roups generate the group G . Notation 6.1 . Let ˙ Φ and ˙ A b e the se ts called Φ and A in 5 .1 and 5.2. W e then set A equal to the set of ﬁx e d p o ints of the automorphism τ of ˙ Φ introduced in 5 .2 (whic h w e con tinue to c a ll τ . Prop ositio n 6. 2. L et ˙ A , ˙ Φ , τ and A b e as in 6.1 and let ¨ α = ˙ α + ˙ α τ (so ¨ α ∈ A sinc e τ 2 = 1 ) for e ach ˙ α ∈ ˙ A . Then the fol lowing hold: (i) If ˙ α ∈ ˙ Φ , then ¨ α 6 = 0 . (ii) L et Φ = { ¨ α/ | ¨ α | | ˙ α ∈ ˙ Φ } . Then Φ is, up to an isometry of A , the r o ot system A 1 in c ases B and G and I 2 (8) (as deﬁne d in 3.1) in c ase F . NON-DISCRETE EUCLIDEAN BUILDINGS 17 (iii) F or e ach ˙ α ∈ ˙ Φ , the half-sp ac e H α := { v ∈ A | v · ¨ α ≥ 0 } of Σ(Φ) (as deﬁn e d in 3.4) is the int erse ction of t he half-sp ac e H ˙ α := { v ∈ ˙ A | v · ˙ α ≥ 0 } of Σ( ˙ Φ) with A . (iv) The map ˙ F 7→ ˙ F ∩ A is an inclus ion-pr eserving bije ction fr om the set of τ -invariant fac es of ˙ Φ to the set of fac es of Φ . Pr o of. In cases B and G , the dimension of ˙ A is only t wo; w e lea ve it to the rea der to verify a ll the claims in these tw o cases . In case F , these asser tions are pr oved in Section 1 .3 o f [11 ]. ✷ Lemma 6.3. L et ˙ Φ and τ b e as in 6.1, let Φ b e as in 6.2.ii, let ˙ ∆ b e an arbitr ary building of typ e ˙ Φ which is not ne c essarily thick, let ρ b e a non-typ e- pr eserving automorphism of ˙ ∆ of or der two and let ˙ Σ b e an ap artment of ˙ ∆ . Then the fol lowing ar e e quivalent: (i) ˙ ψ ◦ τ = ρ ◦ ˙ ψ for some t yp e-pr eserving isomorphism ˙ ψ fr om Σ( ˙ Φ) to ˙ Σ . (ii) ρ ﬁxes two opp osite chamb ers of ˙ Σ . (iii) ρ maps ˙ Σ to itself and ﬁxes at le ast one chamb er of ˙ Σ . Pr o of. Let S b e as in 5.2. W e think of S a s a cham b er of Σ( ˙ Φ) a nd let S ′ be the unique opp os ite cham ber of Σ( ˙ Φ). The map τ ﬁxes b oth S a nd S ′ . Hence if ˙ ψ is as in (i), then ρ ﬁxes the tw o o pp o s ite cham b er s ˙ ψ ( S ) and ˙ ψ ( S ′ ) of ˙ Σ. Thus (i) implies (ii). Since oppo s ite c hambers are contained in a unique apa r tment , (ii) implies (iii). Now supp ose (iii) holds. L e t C b e a chamber o f ˙ Σ ﬁxed by ρ and let ˙ ψ be the unique isomorphism from Σ( ˙ Φ) to ˙ Σ mapping S to C . Since the co mp os ition ρ − 1 ◦ ˙ ψ ◦ τ is also a type-preser ving a utomorphism from Σ( ˙ Φ) to ˙ Σ mapping S to C , it m ust equal ˙ ψ . Thus (i) holds. ✷ Deﬁnition 6.4 . Under the hypotheses of 6.3, we ca ll an apartment o f ˙ ∆ ρ - c omp atible if it sa tisﬁes the three equiv a lent conditio ns in 6.3. In the following result, we make implicit use of the fact that a building ∆ is completely determined by the graph whose vertices are the ch amber s of ∆, where t wo cham b ers are joined by an e dg e whenever ther e is a panel of ∆ co n taining them bo th. (This is the p oint of view, for example, in [14].) Theorem 6 .5. L et ˙ Φ and τ b e as in 6.1, let Φ b e as in 6. 2 .ii, let ˙ ∆ b e an arbitr ary building of typ e ˙ Φ which is not ne c essarily thick, let ρ b e a non-typ e- pr eserving automorphism of ˙ ∆ of or der two and let ˙ Σ b e an ap artment of ˙ ∆ . Supp ose that ˙ Σ is ρ -c omp atible as deﬁne d in 6.4. L et ˙ ψ b e as in 6.3 and let ˙ π b e the bije ct ion fr om ˙ Φ to the set of r o ots of ˙ Σ induc e d by the isomorphism ˙ ψ . L et ∆ b e the gr aph who se vertic es ar e the chamb ers of ˙ ∆ ﬁx e d by ρ , wher e two such chamb ers ar e joine d by an e dge whenever they ar e opp osite in a r esidue of r ank two ﬁ xe d by ρ and let Σ b e the sub gr aph of ∆ sp anne d by the chamb ers of ˙ Σ ﬁ xe d by ρ . Then t he fol lowing hold: (i) Ther e is a unique isomorphism ψ fr om Σ(Φ) t o Σ such that ψ ( ˙ S ∩ A ) = ˙ ψ ( ˙ S ) 18 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS for e ach τ -invariant se ctor ˙ S of ˙ Φ . (ii) If π is the bije ct ion fr om Φ to the set of r o ots of Σ induc e d by ψ , then π ( ¨ α/ | ¨ α | ) = ˙ π ( ˙ α ) ∩ Σ for al l ˙ α ∈ ˙ Φ , wher e t he map ˙ α 7→ ¨ α is as in 6.2 . (iii) ∆ is a building of typ e Φ whose ap artments ar e the sub gr aphs sp anne d by the chamb ers ﬁxe d by ρ in ρ - c omp atible ap artments of ˙ ∆ . Pr o of. Assertions (i) and (ii) hold by 6.2. Next w e observe that if R is the ρ - inv aria n t residue of rank tw o containing t wo a djacent cham b ers of ∆, then all the cham b er s in R ﬁxed by ρ are pair wise opp osite in R and hence a djacent in ∆. This means that ∆ is a chamb er system as deﬁned a t the b eginning of [7 ]. T o show that (iii) holds, it therefore suﬃces , by Theor em 3.11 in [7] and Theore m 8.2 1 in [14], to s how that every tw o cham b ers of ∆ are contained in an a pa rtment o f ˙ ∆ containing o ppo site cham b ers ﬁxed by ρ . In ca ses B and G , every tw o chambers of ∆ ar e opp os ite in ˙ ∆. W e can th us assume that w e are in case F . Let γ = ( C 0 , C 1 , . . . , C k ) b e a gallery in ∆. Then for each i ∈ [1 , k ], the c hambers C i − 1 and C i are opposite in a ρ -inv ariant J i -residue R i , where J i is a tw o-e le men t subset of the vertex se t I of the diag ram F 4 inv aria n t under the non-tr ivial a utomor- phism of this diagr a m. W e will say that γ is alternating if J i 6 = J i − 1 for all i ∈ [2 , k ]. Let ˙ γ b e a gallery in ˙ ∆ from C 0 to C k containing C i also for all i ∈ [1 , k − 1] suc h that the subga llery from C i − 1 to C i is a minimal galler y in R i for each i ∈ [1 , k ]. Thu s the leng th of ˙ γ is m := d 1 + d 2 + · · · + d k , where d i is the diameter of R i for all i ∈ [1 , k ]. Now supp ose that k = 8. Then m equals the dia meter of ˙ Σ and if D a nd D ′ are o pp os ite cham be rs in ˙ Σ ∩ ∆, then there exists a unique minimal g allery from D to D ′ that has the same type as ˙ γ . By Pro po sition 7 .7.ii in [1 4], it follows that ˙ γ is minimal a nd hence C 0 and C 8 are opp osite. By Cor ollaries 8.6 and 8.9 in [14], therefore, ˙ γ is c o ntained in a n apartment of ˙ ∆. It thus suﬃce s to show that ev ery alterna ting g a llery of arbitrary length k in ∆ can b e extended to an a lternating g allery in ∆ of length k + 1 and that every t wo chambers of ∆ ar e joined by an alternating ga llery in ∆. Suppo se ﬁrst that the gallery γ = ( C 0 , C 1 , . . . , C k ) is alternating and let J i for i ∈ [1 , k ] b e as in the previous par agra ph. Let J = I \ J k and let R be the unique J -residue containing C k . Since ρ ﬁxes C k , the r esidue R is a lso ρ -inv a r iant. The residue R is a ge neralized n -gon for n = 2 or 4. Let γ b e a galler y in R of length n/ 2 s tarting a t C k and let γ 1 be the c oncatenation of γ − 1 with γ ρ . Then γ 1 is a minimal ga lley of leng th n (b ecause ρ is not type-preser ving) and ρ preserves γ 1 (beca use ρ 2 = 1). Ther efore ρ ﬁxes the unique cham b er C k +1 opp osite C k in the unique apartment o f R containing γ 1 . Thus ( C 0 , . . . , C k , C k +1 ) is a n alter nating gallery extending γ . Now supp o se that C and C ′ are tw o arbitrar y cham b er s of ∆ and le t e b e the distance be tween them in ˙ ∆. Since ∆ is a cham b er sy s tem (as observed ab ov e), we can obtain an a lternating galler y fr om C to C ′ from an arbitr a ry galler y from C to C ′ simply by disca rding sup erﬂuous chambers. It will s uﬃce to s how, therefore, that ther e is a galler y from C to C ′ in ∆. W e pro c e ed by induction with resp ect to e . W e ca n supp ose that e > 0. Th us we can choos e a cham b er C 1 adjacent to C ′ that is a t distance e − 1 from C . L et R b e the unique ρ -inv ariant residue of rank tw o containing bo th C and C 1 and let C 2 = pro j R C ′ . Since ρ ﬁxes C ′ and NON-DISCRETE EUCLIDEAN BUILDINGS 19 R , it ﬁx e s C 2 to o. Thu s C 2 is a chamber of ∆ adjacent to C in ∆ and at distance strictly less than e to C ′ . By induction, we co nclude that there is, in fact, a galler y in ∆ from C to C ′ . ✷ Notation 6.6. L e t ∆, Σ, ψ , ρ as w ell as U α and x α be as in 5.3. Note that the t yp e o f ∆ is, acco rding to 6 .1, now ˙ Φ r ather than Φ. In order to fo cus on the set ∆ ρ deﬁned in 5.7, we now replac e also the designatio ns ∆, Σ, ψ , U α and x α by ˙ ∆, ˙ Σ, ˙ ψ , U ˙ α and x ˙ α (but let ρ re ma in ρ ). W e then set ∆ = ˙ ∆ ρ and Σ = ˙ Σ ∩ ∆. By 6.5 a pplied to these data, ∆ has (cano nic a lly) the structure of a building o f type Φ, wher e Φ is as in 6.2.ii, and Σ is an apartment of ∆. Let π b e the map obtained from these data in 6.5.ii and for ea ch α ∈ Φ, let ˙ Φ α denote the pre-imag e of α under the surjection ˙ α 7→ ¨ α / | ¨ α | from ˙ Φ to Φ, where the map ˙ α 7→ ¨ α is as in 6.2. Theorem 6.7. L et ˙ Φ , U ˙ α for ˙ α ∈ ˙ Φ , Φ , ˙ Φ α for α ∈ Φ , ∆ , Σ , ρ and π b e as in 6.6 and let Φ b e identiﬁe d with the set of ro ots of Σ via π . F or e ach α ∈ Φ , let U α denote the c entr alizer of ρ in the s u b gr oup h U ˙ α | ˙ α ∈ ˙ Φ α i of Aut( ˙ ∆) . Then U α acts faithful ly on ∆ for e ach α ∈ Φ ( in al l thr e e c ases); (Σ , ( U α ) α ∈ Φ ) is a Moufang struct ur e on ∆ in c ases B and G ; and in c ase F , ∆ is Moufang and for e ach α ∈ Φ , U α is the c orr esp onding r o ot gr oup. Pr o of. Let ˙ ∆ b e as in 6.6 and let α ∈ Φ. W e think of α a s a ro ot of Σ and choose a panel of Σ co nt aining one cham b er C in α a nd a nother C ′ not in α . Then C and C ′ are opp osite in a unique r ank tw o residue ˙ R of ˙ ∆ ﬁxed by ρ . By 6.5.ii, ˙ Φ α consists of precisely those ro o ts o f ˙ Σ that co nt ain C but not C ′ . By Pro po sition 8 .13 in [14], the map ˙ α 7→ ˙ α ∩ ˙ R is thus a bijection from ˙ Φ α to the set of ro ots of the apartment Σ ∩ ˙ R of ˙ R and b y Prop ositio n 11 .10 in [14], U ˙ α induces the ro ot gro up on ˙ R co r resp onding to the ro o t ˙ α ∩ ˙ R for each ˙ α ∈ ˙ Φ α . By Theorem 1 1.11.ii in [14], therefore, the group h U ˙ α | ˙ α ∈ ˙ Φ α i acts sharply transitively o n the se t Q of cham b ers of ˙ R which a re opp osite C in ˙ R . Thu s the gr oup U α acts shar ply tr ansitively o n the ﬁxe d p oint s e t Q ρ of ρ in Q . It follows tha t U α acts faithfully on ∆ and by Theor e m 9.3 in [1 4], that U α acts sharply tr a nsitively on the set of apa rtments of ∆ containing α . W e co nclude that ( U α ) α ∈ Φ is a Moufang structure on ∆ (as deﬁned in 3.8) in cases B and G . Now supp ose that we are in case F and choose a panel of ∆ containing tw o cham b er s C 1 and C ′ 1 in α . There exis ts a unique rank tw o res idue ˙ R 1 of ˙ ∆ ﬁxed by ρ containing C 1 and C ′ 1 . Let ˙ P be a panel o f ˙ R 1 containing tw o chambers o f ˙ Σ. If ˙ α ∈ ˙ Φ α , then ˙ α contains b oth C 1 and C ′ 1 , hence the apa r tment ˙ R 1 ∩ ˙ Σ of ˙ R 1 is co nt ained in ˙ α (since ro ots are conv ex) and thus U ˙ α acts trivially b oth on ˙ R 1 ∩ ˙ Σ and on ˙ P . By Theorem 9.7 in [14], ther e fore, U ˙ α acts trivially o n ˙ R 1 for all ˙ α ∈ ˙ Φ α . It follows that U α is c ontained in the ro ot gro up of ∆ corresp onding to α . T hus ∆ is Moufang since U α acts tra nsitively (in fact, sha rply transitively) on the set o f apartments of ∆ containing α and α is arbitr ary . By Theo rem 9.3 and Prop osition 11.4 in [14], the ro ot group corr esp onding to α a lso ac ts sha rply transitively o n the set of apartments o f ∆ containing α . It follows that U α equals this ro ot g r oup. ✷ 20 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS Con v ent ion 6.8. W e will so metimes r efer to a building ∆ o f the sor t that a ppea rs in 6 .7 as a Su zuki-R e e building . When we say that ∆ is a Suzuki-Ree building in cases B or G , we mean that we hav e in mind the Moufang structure on ∆ descr ibed in 6.7 . Remark 6.9. Let ∆ F and ˙ ∆ F be the buildings called ∆ a nd ˙ ∆ in 6.6 in case F and let ∆ B and ˙ ∆ B denote the buildings c a lled ∆ a nd ˙ ∆ in 6.6 in cas e B under the assumption that L = K , where L is as in 5.1. Then there exist residues of rank tw o of ˙ ∆ F ﬁxed by ρ which are isomor phic to the building ˙ ∆ B . Let ˙ R b e one of these residues a nd let P b e the corres po nding pa nel of ∆ F . Then P can b e identiﬁed with the building ∆ B in such a way that the ca no nical Moufang structure o n P which co mes fro m ∆ F as described in 3 .9 co incides with the Moufang structure on ∆ B describ ed in 6.7. Notation 6.10. Let S = L × L in case B and let S = K × K in case F . In b oth of these tw o cases, let ( s, t ) · ( u, v ) = ( s + u , t + v + s θ u ) and R ( s, t ) = s θ +2 + st + t θ for all ( s, t ) ∈ S . In c a se G , let T = K × K × K , let ( r , s, t ) · ( w , u, v ) = ( r + w, s + u + r θ w, t + v − r u + s w − r θ +1 w ) for all ( r , s, t ) , ( w , u, v ) ∈ T and let N ( r, s, t ) = r θ +1 s θ − r t θ − r θ +3 s − r 2 s 2 + s θ +1 + t 2 − r 2 θ +4 for all ( r , s, t ) ∈ T . Then S and T are groups (with m ultiplication · ), ( s, t ) − 1 = ( s, t + s θ +1 ) for all ( s, t ) ∈ S a nd ( r , s, t ) − 1 = ( − r , − s + r θ +1 , − t ) for all ( r, s, t ) ∈ T . W e w ill call R the norm of the gr oup S and N the norm of the gr oup T . The center of S is { (0 , t ) | t ∈ K (or t ∈ L ) } a nd the c e n ter of T is { (0 , 0 , t ) | t ∈ K } ; b oth centers are isomorphic to the a dditive g roup of K . (Note that in case B , the pr o duct K θ L is contained in L (by 5.1) and thus the pro ducts s θ t , R ( s, t ) 2 − θ u and R ( s, t ) θ v are contained in L for a ll ( s, t ) , ( u, v ) ∈ S even tho ugh the norm R ( s, t ) is not necessar ily contained in L . See 6.12.i.) Remark 6 .11. It is sho wn in [9 ] that the maps R and N are anisotropic. By this we mea n that R ( s, t ) = 0 o nly if ( s, t ) = 0 and N ( r , s, t ) = 0 only if ( r , s, t ) = 0 . Prop ositio n 6.12 . L et ∆ , Φ , Σ , the identiﬁc ation of Φ with the set of r o ots of Σ and the r o ot gr oups U α b e as in 6.7, let G b e the c orr esp onding R e e or Su zuki gr oup as deﬁne d in 5.7 and let t he gr oups S and T and their norms R and N b e as in 6.10. Then ther e exist α ∈ Φ such that t he fol lowing hold: (i) In c ases B and F , U α ∼ = S and ther e exists an isomorphism x α fr om S to U α that x α ( u, v ) h ( s,t ) = x α ( R ( s, t ) 2 − θ u, R ( s, t ) θ v ) for al l ( u, v ) ∈ S and al l ( s, t ) ∈ S ∗ , wher e h ( s, t ) = m Σ ( x α (0 , 1 )) m Σ ( x α ( s, t )) . NON-DISCRETE EUCLIDEAN BUILDINGS 21 (ii) In c ase G , U α ∼ = T and ther e exists an isomorphism x α fr om T to U α such that x α ( w, u, v ) h ( r,s,t ) = x α ( N ( r, s, t ) 2 − θ w, N ( r, s, t ) θ − 1 u, N ( r , s, t ) v ) for al l ( w , u, v ) ∈ T and al l ( r, s, t ) ∈ T ∗ , wher e h ( r , s, t ) = m Σ ( x α (0 , 0 , 1 )) m Σ ( x α ( r , s, t )) . The element α is unique up to the action of the stabilizer G Σ of Σ in G on the set of r o ots of Σ . Pr o of. By [13, 3 3 .17], (i) holds in case F . By 6.9, it follows that (i) holds a lso in case B when K = L . Simply b y r estricting scalars, we conclude that (i) holds in B also when L 6 = K . Suppo se now that we are in c ase G . Let C and C 1 be the tw o cham b ers in Σ a nd suppo se that α = { C } . Let ˙ α i be the element s of ˙ Φ o rdered clo ckwise modulo 12 so that ˙ α 1 , . . . , ˙ α 6 are the ro ots in the set ˙ Φ α deﬁned in 6.7 (wher e ˙ Φ, ˙ α i , etc. are as in 6.6). Let U i denote the r o ot group U ˙ α i of ˙ ∆ for all i , let ( x i ) i ∈ [1 , 6] denote the collection of isomorphisms x i = x ˙ α i from the additiv e group of K to U i which app ear in the rela tions 5.4 – 5 .6 a nd let U + denote the subgr oup of Aut ( ˙ ∆) genera ted by the ro ot gro ups U i for all i ∈ [1 , 6 ]. B y 5 .3.i, x i ( t ) ρ = x 7 − i ( t ) for all i ∈ [1 , 6 ] and by 6.7, U α is the centralizer of ρ in U + . Let (6.13) x α ( r , s, t ) = x 1 ( r ) x 2 ( r θ +1 − s ) x 3 ( t + r s ) x 4 ( r θ +2 − r s + t ) x 5 ( − s ) x 6 ( r ) for all ( r, s, t ) ∈ T . Thu s x α is a ma p from T to U + . By 5.3 – 5.6, 6.10 and a bit of calculation, this map is, in fact, an isomor phism fro m T to U α . Let G † be the subgroup of Aut( ˙ ∆) a s deﬁned in 3 .11. Since the elements in the stabilizer G † C,C 1 are type-pr eserving and ﬁx opp os ite cham b ers of the apa rtment ˙ Σ, they are con tained in the subg roup H of Aut( ˙ ∆) deﬁned in 5.13. If g ∈ G † C,C 1 , then by 5.1 3, there exis t z , z 1 , z 2 ∈ K ∗ such that x α ( r , s, t ) g = x α ( z r, z 1 s, z 2 t ) for all ( r, s, t ) ∈ T . Since x α is a n isomo rphism, the map ( r , s, t ) 7→ ( z r , z 1 s, z 2 t ) is an automorphism of T . It follows that z 1 = z θ +1 and z 2 = z θ +2 t . Thus for each element g of G † ﬁxing the tw o chambers C a nd C 1 of the a partment Σ, there exists an ele ment z ∈ K ∗ such that (6.14) x α ( r , s, t ) g = x α ( z r, z θ +1 s, z θ +2 t ) for all ( r , s, t ) ∈ T . Let the g roup T be identiﬁed with the set ∆ \{ C } via the map sending a ∈ T to the image of C 1 under the element x α ( a ) o f U α . Even though we a re us ing exp onential notation (and, by implication, comp osition from left to right) in the claim we ar e pr oving, for the remainder of this pro o f we will think of the gr o up G as acting on the set C ∪ T from the left (with comp osition from rig h t to left) in order to confor m with the notation in [9] from where we bo rrow the following argument. F or the same reaso n, we will also use additive no tation for T (o nly in this pro of ) even though it is not an a belia n g roup; in par ticular, we let 0 deno te the iden tity (0 , 0 , 0) of T . Thus Σ = { C , 0 } and for ea ch a ∈ T , the elemen t x α ( a ) of U α ﬁxes C and induces the map b 7→ a + b on T . F or each element a = ( r , s, t ) ∈ T ∗ , w e se t (6.15) u ( a ) = r 2 s − rt + s θ − r θ +3 22 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS and (6.16) v ( a ) = r θ s θ − t θ + r s 2 + st − r 2 θ +3 . (These express ions ar e taken from [9, 5.3], wher e N ( a ) is called w = w ( a ). Note that in [9 , 5 .3 ], ther e is an exp onent θ missing in the seco nd term of w and a minus sign mis sing in front of the who le for m ula for w ; see Section 2.10 in [3].) As is explained in Section 5 of [9], N ( a ) 6 = 0 if a 6 = 0 a nd there is a n element ω in G int erchanging the t wo cham b er s C a nd 0 of Σ such that (6.17) ω ( a ) =  − v ( a ) / N ( a ) , − u ( a ) / N ( a ) , − t/ N ( a )  for all a = ( r, s, t ) ∈ T ∗ . By 6.10, 6.1 5, 6 .16 a nd 6.17 (and a bit of ca lc ula tion), we hav e (6.18) N ( ω ( a )) = N ( a ) − 1 and (6.19) N ( − a ) = N ( a ) for all a ∈ T ∗ (where − a is the inverse of a in T ). The elemen t ω 2 ﬁxes 0 and C . B y 6.14, there thus exis ts z ∈ K ∗ such that (6.20) ω 2 ( r , s, t ) = ( z r, z θ +1 s, z θ +2 t ) for all ( r , s, t ) ∈ T . Let v = z θ +2 . Then (0 , 0 , v ) = ω 2 (0 , 0 , 1 ) = ω (1 , 0 , − 1) = (0 , 0 , 1) by 6 .17 and therefore z = v 2 − θ = 1. W e conclude tha t (6.21) ω 2 = 1 by 6 .20. Now let a = ( r , s, t ) ∈ T ∗ and set a ′ = ω ( − ω ( a )). (This makes sense since ω maps T ∗ to itself.) B y 6.2 1, the tw o pro ducts ω x α ( a ) ω a nd x α ( ω ( a )) ω x α ( a ′ ) both map 0 to 0 and C to ω ( a ). Th us (6.22) ρ a :=  x α ( ω ( a )) ω x α ( a ′ )  − 1 ω x α ( a ) ω ﬁxes b o th C and 0. By 6.21 and 6.22, we hav e (6.23) ρ a ( ω ( − a )) = − a ′ = − ω ( − ω ( a )) . Let ξ = ρ a ω . Thus ξ is an ele ment of G interchanging C a nd 0. By 6.2 1 and 6.22, we have ξ = x α ( − a ′ ) · ω x α ( − ω ( a )) ω · x α ( a ) ∈ U ∗ α · U ∗ − α · U ∗ α . By 3 .10.i, therefo r e, ξ = m Σ ( ω x α ( − ω ( a )) ω ) and x α ( a ) = λ ( ω x α ( − ω ( a )) ω ) . Hence by 3 .10.iii, ξ = m Σ ( x α ( a )). Therefor e (6.24) ρ a = m Σ ( x α ( a )) ω . By 6.14, there exists z ∈ K ∗ such that (6.25) ρ a ( w, u, v ) = ( z w , z θ +1 u, z θ +2 v ) NON-DISCRETE EUCLIDEAN BUILDINGS 23 for all ( w, u, v ) ∈ T . By 6.18 and 6.19, we have N ( − ω ( a )) = N ( a ) − 1 . By 6.1 7 a nd 6.25, therefore, the third co or dinate of − ω ( − ω ( a )) is t , wherea s the third co o rdinate of ρ a ( ω ( − a )) is tz θ +2 / N ( a ). By 6 .23, it follows that z θ +2 = N ( a ) and hence (6.26) z = N ( a ) 2 − θ if t 6 = 0 . If t = 0, w e obtain the same conclusion by comparing the ﬁrst or seco nd co ordinates of b oth side s of the identit y 6.23; we leav e these calculatio ns to the reader. By 6.25 and 6 .26, ﬁnally , w e have ρ a = 1 if a = (0 , 0 , 1). By 6.2 1 a nd 6.24, it follows that ω = m Σ (0 , 0 , 1 ). Thus (ii) holds by 6.2 4, 6 .25 and 6.2 6. (Note that in (ii), x α ( w, u, v ) h ( r,s,t ) is to be interpreted a s x α ( w, u, v ) conjugated ﬁrst by m Σ ( x α (0 , 0 , 1 )) and then by m Σ ( x α ( r , s, t )), wherea s in the pro of ρ a ( w, u, v ) is to be in terpreted as the image of ( w , u, v ) under m Σ ( x α (0 , 0 , 1 )) to which then the map m Σ ( x α ( r , s, t )) is a pplied.) ✷ 7. Bruha t-Tits sp aces f or the Ree a nd S uzuki gr oups W e beg in this sectio n with a result which explains why a v aluation of the ro ot datum of a Suzuk i-Ree building deﬁned ov er a pair ( K, θ ) (or triple ( K, θ , L )) requires the existence of a θ -inv a riant v aluatio n of K . W e then formulate o ur mo st impo rtant r esult in 7.11. Theorem 7.1. L et ∆ , Σ , Φ , α , x α , et c. b e as in 6.6 and 6.12, let w = x α (0 , 1 ) in c ases B and F and let w = x α (0 , 0 , 1 ) in c ase G , let ψ b e a valuation of the r o ot datum of ∆ b ase d at Σ and let ϕ = ψ − ψ α ( w ) α/ ( α · α ) (as deﬁne d in 3.15). (Thus ϕ is a valuation e quip ol lent to ψ such that ϕ α ( w ) = 0 .) Then ther e exists a u nique θ -invariant valuation ν of K , which dep ends only on the e quip ol lenc e class of ψ , su ch that (7.2) ϕ α ( x α ( s, t )) = ν  R ( s, t )  for al l ( s, t ) ∈ S ∗ in c ases B and F and (7.3) ϕ α ( x α ( r , s, t )) = ν  N ( r, s, t )  for al l ( r , s, t ) ∈ T ∗ in c ase G . Pr o of. Let ν ( t ) = ϕ α ( x α (0 , t θ )) / 2 for all t ∈ K ∗ in cases B and F and le t ν ( t ) = ϕ α ( x α (0 , 0 , t )) / 2 for all t ∈ K ∗ in cas e G . Then 7.2 a nd 7.3 hold by 3.18 with g = w and u = x α ( s, t ) or x α ( r , s, t ) and 6.12. It th us need only to show that ν is a θ -inv ar iant v aluation of K . Let ν (0) = ∞ . By (V1), we hav e ν ( s + t ) ≥ min { ν ( s ) , ν ( t ) } for all s, t ∈ K . By 6.12 a gain, we have x α (0 , s θ ) m Σ ( w ) m Σ ( x α (0 ,t θ )) = x α (0 , s θ t 2 θ ) for all s, t ∈ K ∗ in cas es B and F and x α (0 , 0 , s ) m Σ ( w ) m Σ ( x α (0 , 0 ,t )) = x α (0 , 0 , st 2 ) 24 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS for all s , t ∈ K ∗ in cas e G . By 3.18 ag ain, this time with g = x α (0 , t θ ) or x α (0 , 0 , t ) and u = x α (0 , s θ ) or x α (0 , 0 , s ), it follows that ν ( s 2 t ) = ν ( t ) + 2 ν ( s ) for all s, t ∈ K ∗ in all three cases. ¿F rom this identit y , we thus obtain ν ( s 2 t 2 ) = ν ( t 2 ) + 2 ν ( s ) for all s, t ∈ K ∗ and (setting t = 1) ν ( s 2 ) = 2 ν ( s ) for all s ∈ K ∗ . Therefo re ν ( st ) = ν ( s ) + ν ( t ) for all s, t ∈ K ∗ . Since | ϕ α ( U ∗ α ) | > 1 b y 3.1 4, it follows from 7.2 and 7.3 that | ν ( K ∗ ) | > 1 . Thus ν is a v aluatio n of K . It remains only to show that ν ( u ) ≥ 0 implies that ν ( u θ ) ≥ 0. Let u b e an element of K ∗ such that ν ( u ) ≥ 0 (and hence ν ( u 2 ) = 2 ν ( u ) ≥ 0). Supp ose ﬁrst that we a re in case B or F , s o (1 , 0 ) · (0 , u θ ) = (1 , u θ ) in S . By (V1), ther efore, ϕ α ( x α (1 , u θ )) ≥ min { ϕ α ( x α (1 , 0 )) , ϕ α ( x α (0 , u θ )) } . Hence by 7 .2, ν (1 + u θ + u 2 ) ≥ min { 0 , ν ( u 2 ) } = 0 . It follo ws that ν ( u θ ) ≥ 0 . Suppo se no w that we are in case G , s o (1 , 0 , 0 ) · (0 , 0 , u ) = (1 , 0 , u ) in T . By (V1), therefore, ϕ α ( x α (1 , 0 , u )) ≥ min { ϕ α ( x α (1 , 0 , 0 )) , ϕ α ( x α (0 , 0 , u )) } . Hence by 7 .3, ν ( u 2 − u θ − 1 ) ≥ min { 0 , ν ( u 2 ) } = 0 . Again we co nclude that ν ( u θ ) ≥ 0 . ✷ The con verse o f 7.1 is also v alid: Theorem 7.4. L et ∆ , Σ , Φ , α , x α , et c. b e as in 6.6 and 6.12. S upp ose t hat ν is a θ - invariant valuation of K and let ϕ α b e the map given in 7.2 (in c ases B and F ) or 7.3 (in c ase G ). Then ϕ α extends to a valuation of t he r o ot datum of ∆ b ase d at Σ . It would not b e har d to prove this res ult directly . The principal diﬃcult y is to show that (7.5) ν  R  ( s, t ) · ( u, v )  ≥ min  ν  R ( s, t )  , ν  R ( u, v )  for all ( s, t ) , ( u, v ) ∈ S and (7.6) ν  N  ( r , s, t ) · ( w , u, v )  ≥ min  ν  N ( r, s, t )  , ν  N ( w , u, v )  for all ( r , s, t ) , ( w , u, v ) ∈ T . These inequalities are requir ed to verify (V1). Rather than prove 7 .4 dire ctly , how ever, we will prove a str o nger result (7.7 – 7.11) which will hav e 7.4 (and thus also the t wo inequalities 7 .5 and 7.6) as corolla ries. NON-DISCRETE EUCLIDEAN BUILDINGS 25 In Section 9 we include a direct pr o of of the inequa lities 7.5 and 7 .6 only b ecause it migh t b e of some indep endent interest. See also 9.1 .10 in [2]. Notation 7.7. Let ˙ Φ, ˙ A , A a nd τ b e a s in 6.1, let ˙ ∆, ∆, ˙ Σ and Σ b e as in 6.6, let ˙ W b e the W eyl g r oup of ˙ Φ, let W be the res triction of the centralizer of τ in ˙ W to the subspac e A and let W be the group of all isometries of A generated by W and all translatio ns o f A . Let ν b e a θ -inv ar iant v a lua tion of K , let ρ b e the automorphism of ˙ ∆ describ ed in 5 .3.i, let ˙ Φ b e identiﬁed with the s e t of ro ots of ˙ Σ via the map called ˙ π in 6.5 a nd let ˙ ϕ denote the ρ -inv a r iant v alua tion of the ro ot datum o f ˙ ∆ based a t ˙ Σ deter mined by ν a s des c rib ed in 5.15. Let ( ˙ X , ˙ A ) be the Bruhat-Tits space o f type ˙ Φ, ˙ A the apartment of ( ˙ X , ˙ A ) and x A the p oint of ˙ A obtained by a pplying 4.16 to ˙ ∆, ˙ Σ and ˙ ϕ . The pair ( ˙ X , ˙ A τ ) deﬁned as in 4.4 can also b e thought o f as the B r uhat-Tits spac e o f type ˙ Φ obta ined by applying 4 .16 to ˙ ∆, ˙ Σ and ˙ ϕ , but only after identifying ˙ Φ with the se t of r o ots of ˙ Σ via ˙ π ◦ τ rather than ˙ π . By the uniqueness a ssertion in 4.16, there exists a τ -automor phism ˙ ρ of ( ˙ X , ˙ A ) (as de ﬁned in 4 .4) that induces the automorphism ρ on ˙ ∆ and maps the pair ( ˙ A, x A ) to itself. This map satisﬁes (7.8) ˙ ρ ◦ ˙ f ˙ A,x A = ˙ f ˙ A,x A ◦ τ , where ˙ f ˙ A ,x A ∈ ˙ A is a s deﬁned in 4.8. Let ˙ A ρ denote the s e t of charts ˙ f ∈ ˙ A such that ˙ ρ maps the apartmen t ˙ f ( ˙ A ) to itself and a cts triv ia lly o n ˙ f ( A ), let A = { ˙ f | A | ˙ f ∈ ˙ A ρ } , and let (7.9) X = [ ˙ f ∈ ˙ A ρ ˙ f ( A ) . Thu s X is contained in the ﬁxed p oint set ˙ X ˙ ρ of ˙ ρ . By 7.8, ˙ f ˙ A,x A ∈ ˙ A ρ and hence (7.10) A := ˙ f ˙ A,x A ( A ) is a subset of X and x A = ˙ f ˙ A,x A (0) is a po in t of A . Here now is our main res ult: Theorem 7 .11. L et Φ , ν , ( X , A ) , A and x A b e as in 7.7. Thus, in p articular, ν is a θ -invariant valuation of K . Then the fol lowing hold: (i) The p air ( X, A ) is a Bruhat-Tits sp ac e of typ e Φ whose building at inﬁnity is ∆ (in the sen s e of 4.14 if t he r ank of ∆ is one) and A is an ap artment of ( X , A ) . (ii) L et α ∈ Φ b e as in 6.12 and let ϕ b e the valuation of the r o ot datum of ∆ b ase d at Σ that app e ars in 4.12.ii when 4.12 (and then 4.15) is applie d to the triple ( X, A ) , A and x A . Then ther e exists a valuation ν 1 e quivalent to ν such that ϕ α satisﬁes 7.2 or 7.3 with ν 1 in plac e of ν . Note that 7.4 is a consequence o f 7.11. 8. The Proof of 7.11 F or the rest of this pap er, we let ˙ ∆, ∆, ( ˙ X , ˙ A ), ˙ Φ, ( ˙ A , ˙ W ), τ , ρ , Φ, ( A , W ), ( X, A ), A , x A , ˙ ρ , ˙ f ρ , etc. b e as in 7.7. Let ˙ X ˙ ρ denote the set o f ﬁxed p oints o f ˙ ρ . W e prov e 7.11 in a ser ies of steps. 26 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS Prop ositio n 8.1. Supp ose that ˙ A 1 is an ap artment of ( ˙ X , ˙ A ) . Then the fol lowing ar e e quivalent: (i) ˙ A 1 is ˙ ρ - invariant and c ontains at le ast one se ctor ﬁxe d by ˙ ρ . (ii) Ther e exists a chart ˙ f 1 in ˙ A ρ such that ˙ ρ ◦ ˙ f 1 = ˙ f 1 ◦ τ . (iii) ˙ A 1 is the image of a chart in ˙ A ρ . If ˙ f 1 is as in (ii), then ˙ X ˙ ρ ∩ ˙ A 1 = X ∩ ˙ A 1 = ˙ f 1 ( A ) . Pr o of. Let ˙ f b e a chart in ˙ A suc h that ˙ A 1 = ˙ f ( ˙ A ). Supp ose ﬁrst that ˙ A 1 is ˙ ρ -inv ariant a nd that there ex ists a sector S of ˙ Φ suc h that (8.2) ˙ ρ ( ˙ f ( S )) = ˙ f ( S ) . By 4.4, there exists ˙ f ′ ∈ ˙ A s uch that ˙ ρ ◦ ˙ f = ˙ f ′ ◦ τ . Since ˙ A 1 is ˙ ρ -inv aria nt, also ˙ f ′ ( ˙ A ) equals ˙ A 1 . By (A2) (in 4.2), therefo r e, ther e exis ts ˙ w ∈ ˙ W such that ˙ f ′ = ˙ f ◦ ˙ w . Thus (8.3) ˙ ρ ◦ ˙ f = ˙ f ◦ ˙ w ◦ τ . By 8.2, it follows that ˙ w ◦ τ ﬁxes S . Thus ˙ w ◦ τ is a non-trivia l automorphism of Σ( ˙ Φ) ﬁxing S . Ther e is only o ne such automor phis m. Since τ also ﬁxes a s ector of ˙ Φ and ˙ W acts transitively on the set o f sectors o f ˙ Φ, it fo llows that there exists ˙ w 1 ∈ ˙ W such that ˙ w ◦ τ = ˙ w 1 ◦ τ ◦ ˙ w − 1 1 . Let ˙ f 1 = ˙ f ◦ ˙ w 1 . Then ˙ f 1 ∈ ˙ A by (A1) and ˙ ρ ◦ ˙ f 1 = ˙ ρ ◦ ˙ f ◦ ˙ w 1 = ˙ f ◦ ˙ w ◦ τ ◦ ˙ w 1 = ˙ f ◦ ˙ w 1 ◦ τ = ˙ f 1 ◦ τ by 8 .3. It follows from this iden tit y that ˙ ρ acts trivia lly on ˙ f 1 ( A ), s o ˙ f 1 ∈ ˙ A ρ (but ˙ ρ do es not ﬁx a n y other po int s in ˙ A 1 , so ˙ X ˙ ρ ∩ ˙ A 1 = X ∩ ˙ A 1 = ˙ f 1 ( A )). Thus (i) implies (ii). It now suﬃces to observe that if ˙ f 1 is a chart in ˙ A ρ whose imag e is ˙ A 1 , then b y 6.2.iv, ˙ ρ ﬁxes ˙ f 1 ( S ) for every τ -inv ariant sector S of ˙ Φ. ✷ Prop ositio n 8. 4. L et ˙ f and ˙ f 1 b e two charts in ˙ A ρ such that ˙ A 1 := ˙ f ( ˙ A ) = ˙ f 1 ( ˙ A ) . Then ˙ f ( A ) = ˙ f 1 ( A ) = ˙ X ˙ ρ ∩ ˙ A 1 = X ∩ ˙ A 1 . Pr o of. By 8.1, it suﬃces to assume tha t ˙ ρ ◦ ˙ f 1 = ˙ f 1 ◦ τ and ˙ X ˙ ρ ∩ ˙ A 1 = X ∩ ˙ A 1 = ˙ f 1 ( A ). Thu s ˙ f ( A ) ⊂ ˙ X ˙ ρ ∩ ˙ A 1 = ˙ f 1 ( A ). By (A2), ther efore, there exists ˙ w ∈ ˙ W such that ˙ f 1 ◦ ˙ w and ˙ f co incide on A . Thus ˙ f 1 ( ˙ w ( A )) = ˙ f ( A ) ⊂ ˙ f 1 ( A ) and hence ˙ w maps A to itself. Therefore ˙ f 1 ( A ) = ˙ f ( A ). ✷ Prop ositio n 8.5. L et x ∈ X and let g x b e as in 4.10. Then the set of ﬁxe d p oints of ˙ ρ in t he r esidue ( ˙ X , ˙ A ) x has the structur e of a building of typ e Φ and for al l ˙ f ∈ ˙ A ρ mapping 0 t o x , g x ( ˙ f ( A )) is an ap artment of this building. Pr o of. By 4.10, ( ˙ X , ˙ A ) x is a building of type ˙ Φ whos e apartments are all of the form g x ( ˙ f ( ˙ A )) for some ˙ f ∈ ˙ A . The co nc lus ion ho lds, there fo re, by applying 6.5 to this building and the a utomorphism of this building induced by ˙ ρ . ✷ Corollary 8 .6. L et x ∈ X , let ˙ u b e a germ at x ﬁxe d by ˙ ρ and let ˙ f b e a chart in ˙ A ρ mapping 0 to x . Then ther e exists a se ctor with vertex x in t he ap artment g x ( ˙ f ( ˙ A )) that is ﬁxe d by ˙ ρ and whose germ is opp osite a maximal germ at x c ontaining ˙ u . NON-DISCRETE EUCLIDEAN BUILDINGS 27 Pr o of. The germ ˙ u is a face and g x ( ˙ f ( A )) is an apartment of the building of type Φ describ ed in 8.5. Since the apartmen t g x ( ˙ f ( ˙ A )) is ˙ ρ -in v ar ia nt , a sector with vertex x in this a partment is ﬁxed by ˙ ρ if and only if its germ is ﬁx ed by ˙ ρ . The claim follows, therefo re, from the fact that for each cham b er C and eac h apartment Σ in a s pherical building, there alwa ys exists a cham b er in Σ whic h is opp os ite C . ✷ Prop ositio n 8.7. L et ˙ A 1 b e t he image of a chart ˙ f 1 in ˙ A ρ . Then the map ˙ S 1 7→ ˙ S 1 ∩ X is a bije ction fr om the set of se ctors of ˙ A 1 that ar e ﬁ x e d by ˙ ρ to t he set of se ctors of ˙ f 1 ( A ) , i.e. to the set of images under ˙ f 1 of se ctors of Φ . Pr o of. By 6.2, the map ˙ S 7→ ˙ S ∩ A is a bijection fro m the set of sectors of ˙ Φ that are ﬁxed by τ to the s e t of s ectors of Φ. If ˙ S is any one of these s ectors, then ˙ f 1 ( ˙ S ∩ A ) = ˙ f 1 ( ˙ S ) ∩ X by 8 .4. ✷ Prop ositio n 8.8. The p air ( X , A ) is a non-discr ete Euclide an bu ilding of typ e Φ . Pr o of. W e need to show that ( X, A ) satisﬁes the conditions (A1)–(A6) form ulated in 4.2 . Let f ∈ A and w ∈ W . Ther e exist element s ˙ f in ˙ A ρ and ˙ w in the centralizer of τ in ˙ W such that f is the restr iction of ˙ f to A and w is the restriction of ˙ w to A . Since ( ˙ X , ˙ A ) satisﬁes (A1), we have ˙ f ◦ ˙ w ∈ ˙ A . It follo ws that f ◦ w ∈ ˙ A ρ since ˙ f ( ˙ w ( ˙ A )) = ˙ f ( ˙ A ) and ˙ f ( ˙ w ( A )) = ˙ f ( A ). Thus ( X , A ) satisﬁes (A1). Next let ˙ f , ˙ f ′ ∈ ˙ A ρ . Since ( ˙ X , ˙ A ) s atisﬁes (A2), the set ˙ M := { v ∈ ˙ A | ˙ f ( v ) ∈ ˙ f ′ ( ˙ A ) } is closed a nd conv ex and there exists ˙ w ∈ ˙ W such that the maps ˙ f and ˙ f ′ ◦ ˙ w coincide on ˙ M . Let M := { v ∈ A | ˙ f ( v ) ∈ ˙ f ′ ( A ) } . Then M ⊂ ˙ M ∩ A . Let v ∈ ˙ M ∩ A . Thus v ∈ A and ˙ f ( v ) = ˙ f ′ ( v ′ ) for some v ′ ∈ ˙ A . Since ˙ f ∈ ˙ A ρ , the p oint ˙ f ( v ) is a po in t o f ˙ f ′ ( ˙ A ) ﬁxed by ˙ ρ . Since als o ˙ f ′ ∈ ˙ A ρ , it follows by 8.4 that v ′ ∈ A and hence v ∈ M . W e conclude that M = ˙ M ∩ A . Since ˙ M is closed and conv ex, it follows that M is also clos ed and conv ex . T o ﬁnish showing that ( X , A ) satisﬁes (A2), we ca n a s sume that | M | > 1. By (A1), we can assume further that the origin 0 is contained in M and that ˙ w ﬁxes 0, s o ˙ w ∈ ˙ W . Let ˙ f ∗ and ˙ f ′ ∗ be as in 4 .11 and let Ξ denote the building of type Φ described in 8.5. B y 6.2.iv, we ca n think o f the τ -in v ariant faces o f Σ( ˙ Φ) as the faces of Σ(Φ). Thus ˙ f ∗ and ˙ f ′ ∗ bo th map Σ(Φ) to apa rtments of the building Ξ. In cases B and G , a n apartment of Ξ is a n arbitrary tw o-element set of cham b ers . In case F , an a partment of Ξ is a circuit consisting of 16 chambers a nd 16 panels and tw o distinct apar tmen ts intersect either in a connected piece of this circuit (po ssibly empty) o r in tw o opp os ite panels . Let Y b e the set of all faces of Σ(Φ) which ar e ma pp ed by ˙ f ∗ to ˙ f ′ ∗ (Σ(Φ)). Thus either Y = Σ(Φ), Y is a simplicial arc in Σ(Φ) (p ossibly empty) o r we are in case F and Y consists of tw o opp os ite panels (i.e. tw o opp osite non-maxima l face s). The map ( ˙ f ′ ∗ ) − 1 ◦ ˙ f ∗ from Y into Σ(Φ) is t yp e-pres erving (since the maps ˙ f ∗ and ˙ f ′ ∗ are type-preser ving) and if Y consists o f t wo opp os ite panels of a given type, then ( ˙ f ′ ∗ ) − 1  ˙ f ∗ ( Y )  consists o f tw o opp osite panels of the same type (since the maps ˙ f ∗ and ˙ f ′ ∗ bo th map opp osite panels to 28 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS opp osite panels). Th us in every case there exists a n element ˙ w 1 in the centralizer of τ in ˙ W such that ˙ f ∗ and ˙ f ′ ∗ ◦ ˙ w 1 coincide on Y . Let z b e an a rbitrary non- zero element o f M and let z ′ = ˙ w ( z ). Since M is conv ex, it contains the closed interv al [0 , z ]. Let ˙ F be the unique minimal face of Σ(Φ) that co ntains [0 , z ]. Since ˙ f a nd ˙ f ′ ◦ ˙ w coincide on M , we have z ′ ∈ A and ˙ f ( z ) = ˙ f ′ ( z ′ ). If ˙ F ′ is the minimal face o f Σ(Φ) tha t contains [0 , z ′ ], then ˙ f ( ˙ F ) = ˙ f ′ ( ˙ F ′ ). Ther efore ˙ F ∈ Y (and, in particular , Y is not empty). Thus ˙ f ′ ∗ ( ˙ F ′ ) = ˙ f ∗ ( ˙ F ) = ˙ f ′ ∗ ( ˙ w 1 ( ˙ F )) by the conclusion of the previous para graph and hence ˙ F ′ = ˙ w 1 ( ˙ F ). It follows that ˙ w ( z ) = z ′ ∈ ˙ w 1 ( ˙ F ), so the p oints z a nd ˙ w − 1 1 ˙ w ( z ) are b oth co nt ained in the face ˙ F . Since every p oint of ˙ A distinct fro m the orig in is contained in a t most one fa ce of ˙ Φ of a given type and the sta bilizer of a face in ˙ W acts trivially on that face, it follows that ˙ w 1 ( z ) = ˙ w ( z ). W e conclude that ˙ f ′ ◦ ˙ w 1 coincides with ˙ f on M . Thus ( X , A ) satisﬁes (A2). Now let x, x ′ ∈ X . Since ( ˙ X , ˙ A ) satisﬁes (A3), there exis ts an apa rtment ˙ A 1 of ( ˙ X , ˙ A ) co nt aining the interv al [ x, x ′ ] (as deﬁned in 4.6). By 7.9, there ex ist ˙ f , ˙ f ′ ∈ ˙ A ρ such that x ∈ ˙ f ( A ) and x ′ ∈ ˙ f ′ ( A ). Since ˙ ρ ﬁxes x and x ′ , it ﬁxes the po int g x ( x ′ ) of the r esidue ( ˙ X , ˙ A ) x deﬁned in 4.10. Let ˙ F be the unique minimal face of the apartment ˙ A 1 with v ertex x whose germ contains the point g x ( x ′ ) and let ˙ u denote the germ o f ˙ F . The germ ˙ u is ﬁxed b y ˙ ρ (since other wise ˙ u ˙ ρ would b e disjoint from ˙ u ). By 8.6, the apartment g x ( ˙ f ( ˙ A )) contains a sector with vertex x which is b oth ﬁxed by ˙ ρ and whose germ is o ppo site a maximal g erm a t x containing ˙ u . By Lemma 1.13 in [6 ], there exists an a partment ˙ A 2 containing ˙ S 1 ∪ ˙ F . Th us [ x, x ′ ] ⊂ ˙ F ⊂ ˙ A 2 . The conv ex closure of ˙ S 1 ∪ { x ′ } is a sector of ˙ A 2 with v ertex x ′ which contains the interv al [ x, x ′ ]. W e denote this se ctor by ˙ S 2 . Since ˙ ρ ﬁxes ˙ S 1 and x ′ , it ﬁxes ˙ S 2 as w ell. By a seco nd application o f 8.6, there exists a sector ˙ S 3 with vertex x ′ in the apartment ˙ f ′ ( ˙ A ) that is ﬁxed by ˙ ρ and whose germ is opp osite the germ of ˙ S 2 at x ′ . B y Lemma 1 .1 2 in [6], there exists a unique apar tment ˙ A 3 containing ˙ S 2 ∪ ˙ S 3 . This apartment con tains [ x, x ′ ] (since ˙ S 2 contains [ x, x ′ ]) and is ˙ ρ -in v ar iant (since ˙ S 2 and ˙ S 3 are ˙ ρ -in v ar iant). By 8 .1, there exists a chart ˙ f 1 in ˙ A ρ such that ˙ A 3 = ˙ f 1 ( ˙ A ) and x, x ′ ∈ ˙ f 1 ( A ). Thus ( X , A ) satisﬁes (A3). Let S and S ′ be tw o sectors of X . By 8.7, there exist ˙ ρ -inv ariant sector s ˙ S and ˙ S ′ of ( ˙ X , ˙ A ) suc h that S = ˙ S ∩ X and S ′ = ˙ S ′ ∩ X . The chambers ˙ S ∞ and ( ˙ S ′ ) ∞ of ( ˙ X , ˙ A ) ∞ are contained in a ρ -inv ar iant apar tment o f ( ˙ X , ˙ A ) ∞ . This a pa rtment is o f the form ˙ A ∞ 1 , where ˙ A 1 is a ˙ ρ -in v ar iant apar tmen t of ( ˙ X , ˙ A ). The apartment ˙ A 1 contains sec to rs ˙ S 1 and ˙ S ′ 1 such that ˙ S 1 ⊂ ˙ S and ˙ S ′ 1 ⊂ ˙ S ′ . Let ˙ S 2 = ˙ S 1 ∩ ˙ ρ ( ˙ S 1 ) and ˙ S ′ 2 = ˙ S ′ 1 ∩ ˙ ρ ( ˙ S ′ 1 ). The intersection of tw o subsectors of a g iven secto r is aga in a subsector. It follows that ˙ S 2 is a s ubsector of ˙ S and ˙ S ′ 2 is a subsector of ˙ S ′ . Since ˙ ρ 2 = 1 , b oth of these subsectors a re ﬁxed by ˙ ρ . By 8.1, therefore , ˙ A 1 = ˙ f 1 ( ˙ A ) for some ˙ f 1 ∈ ˙ A ρ . Hence by 8.7, ˙ S 2 ∩ X and ˙ S ′ 2 ∩ X ar e subsectors of S and S ′ bo th are co n tained in ˙ f 1 ( A ). Thus ( X , A ) satisﬁes (A4). W e turn now to (A5). A roo t of ( X, A ) is the image under a chart in ˙ A ρ of H α for some α ∈ Φ, where H α is a s in 3.4. Let ˙ A 1 and ˙ A 2 be the images of tw o c harts in ˙ A ρ such that ˙ A 1 ∩ ˙ A 2 ∩ X contains a ro ot β o f ( X , A ) but ˙ A 1 ∩ X 6 = ˙ A 2 ∩ X . By (A2), the set ˙ A 1 ∩ ˙ A 2 ∩ X is clo sed. W e can therefore choose a po int x in this set such that the apar tmen ts ˙ Σ 1 := g x ( ˙ A 1 ) and ˙ Σ 2 := g x ( ˙ A 2 ) of the res idue of ( ˙ X , ˙ A ) x are distinct. F or i = 1 and 2, let ˙ Σ ˙ ρ i be the set of c hambers of ˙ Σ i ﬁxed by ˙ ρ . Then NON-DISCRETE EUCLIDEAN BUILDINGS 29 ˙ Σ ˙ ρ 1 and ˙ Σ ˙ ρ 2 bo th span apartmen ts of the building Ξ of type Φ deﬁned in 8.5. Let S be a se ctor contained in the ro ot β . Then the conv e x hull S ′ of { x } ∪ S is a sector with v ertex x a nd ˙ Σ ˙ ρ 1 and ˙ Σ ˙ ρ 2 bo th con tain the unique cham b er of Ξ that c o nt ains g x ( S ′ ). It follows that ˙ Σ ˙ ρ 1 ∩ ˙ Σ ˙ ρ 2 contains a ro o t of b o th ˙ Σ ˙ ρ 1 and ˙ Σ ˙ ρ 2 . On the other hand, ˙ Σ ˙ ρ 1 6 = ˙ Σ ˙ ρ 2 since otherwis e ˙ Σ 1 ∩ ˙ Σ 2 would contain a pair of opp osite c hambers. Two distinct apartments of a spher ical building whose in ter section contains a roo t int erse c t in a ro ot and their symmetric diﬀerence spans a third apar tment . Thu s the s y mmetric diﬀerence of ˙ Σ ˙ ρ 1 and ˙ Σ ˙ ρ 2 spans a third apartment of Ξ. Let ˙ u 1 and ˙ u 2 be tw o cham b ers in this apartment that are o ppo site in Ξ and hence also opposite in the r esidue ( X , A ) x . Ther e exist unique sector s ˙ S 1 and ˙ S 2 of ˙ A 1 and ˙ A 2 with vertex x whose germs a re ˙ u 1 and ˙ u 2 , and b y Pr op osition 1.12 in [6] there exis ts an apartment ˙ A 3 containing these t wo sectors . In pa rticular, x ∈ ˙ A 1 ∩ ˙ A 2 ∩ ˙ A 3 ∩ X . Now s upp os e that ˙ A ′ is the imag e of a c hart ˙ f ′ in ˙ A ρ such that both ˙ A ′ ∩ ˙ A 1 ∩ X and ˙ A ′ ∩ ˙ A 2 ∩ X co n tain ro ots of ( X , A ) that ar e disjoint from ˙ A 1 ∩ ˙ A 2 ∩ X . By 8.7, ˙ f ′ ( A ) contains subsector s of b oth ˙ S 1 ∩ X a nd ˙ S 2 ∩ X and hence ˙ A ′ contains subsectors of b oth ˙ S 1 and ˙ S 2 . Thus ˙ A ′ = ˙ A 3 since ˙ A 3 is the conv ex hull of a ny tw o sectors, one contained in S 1 and the other in S 2 . Therefore x ∈ ˙ A 1 ∩ ˙ A 2 ∩ ˙ A ′ . Thus ( X, A ) satisﬁes (A5). Let ˙ d b e the metric on ˙ X that app ear s in (A6). The restriction of ˙ d to X is a metric o n X , a nd each chart in A is the restriction to A of a c hart in ˙ A . Therefor e ( X, A ) satisﬁes (A6) with the r estriction o f ˙ d to X in place o f d . ✷ Prop ositio n 8. 9. The building ∆ is the building at inﬁ nity of ( X, A ) (in the sense of 4.14 in c ases B and G ). Pr o of. By 8.7, there is a canonical bijection π from the cham b er s e t of ( X , A ) ∞ to the set of all c hambers ˙ S ∞ of ( ˙ X , ˙ A ) ∞ = ˙ ∆ such that ˙ S is a sector o f ( ˙ X , ˙ A ) tha t is ﬁxe d by ˙ ρ . If ˙ S 1 is a n ar bitrary sector of ( ˙ X , ˙ A ) such that ˙ S ∞ 1 is ﬁxe d b y ρ , then ˙ S 2 := ˙ S 1 ∩ ˙ S ˙ ρ 1 is a s ector of ( ˙ X , ˙ A ) ﬁxed by ˙ ρ suc h that ˙ S ∞ 2 = ˙ S ∞ 1 . It follo ws that π is, in fact, an is omorphism from ( X , A ) ∞ to ∆. F urthermore, the conclusions of 4.12 hold for ( X, A ) and ∆, even in case s B and G , by 4 .12 applied to ( ˙ X , ˙ A ) and 6.7. ✷ By 8 .8 and 8 .9, we co nclude that 7.11.i holds. Now let α ∈ Φ and ϕ b e as in 7.11.ii. Prop ositio n 8.10. Ther e exist s a p ositive r e al numb er k (which dep ends on t he c ase) su ch that ϕ α ( x α (0 , t )) = k · ν ( t ) for al l t ∈ K ∗ (or t ∈ L ∗ ) in c ase B or F and ϕ α ( x α (0 , 0 , t )) = k · ν ( t ) for al l t ∈ K ∗ in c ase G . Pr o of. Suppo se ﬁrst that w e are in case G . W e use the notation from the pro o f of 6.12. By 6.13, in particular , we hav e x α (0 , 0 , t ) = x ˙ α 4 ( t ) x ˙ α 5 ( t ) for all t ∈ K . Since τ int erchanges ˙ α 4 and ˙ α 5 , there exists a p ositive r e al num b er k such that for a ll t ∈ K ∗ , the aﬃne half-spaces H ˙ α 4 ,ν ( t ) and H ˙ α 5 ,ν ( t ) (as deﬁned in 30 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS 4.12) b oth in tersect A (which is the space o f ﬁxed p o ints of τ ) in a n aﬃne half-spa ce of the fo r m H α,k · ν ( t ) . Cho ose t ∈ K ∗ , let u = x ˙ α 4 ( t ), let y = x ˙ α 5 ( t ) and let z = u y . Then ˙ A ∩ ˙ A u = ˙ f ˙ A ,x A ( H ˙ α 4 ,ν ( t ) ) and ˙ A ∩ ˙ A y = ˙ f ˙ A,x A ( H ˙ α 5 ,ν ( t ) ) by 4 .12.ii and 5.15. Supp ose that z ﬁxes a p oint x in A which is not in ˙ f ˙ A,x A ( H α,k · ν ( t ) ) . Cho ose x ′ in ˙ f ˙ A,x A  H ˙ α 4 ,ν ( t ) ∩ H ˙ α 5 ,ν ( t )  but not in ˙ f ˙ A,x A ( A ). Since u and y both ﬁx x ′ , so do es z . Thus z ﬁxes every po int in the interv al [ x, x ′ ]. This interv al contains p oints, how ever, which are in ˙ f ˙ A,x A ( H ˙ α 4 ,ν ( t ) ) or ˙ f ˙ A,x A ( H ˙ α 5 ,ν ( t ) ) but not in bo th. These p oints are ﬁxed b y u or y but not both. Hence they cannot b e ﬁxed by z . W e conclude that A ∩ A z = ˙ f ˙ A,x A ( H α,k · ν ( t ) ) . By 4.8, f A,x A is the restriction of ˙ f ˙ A,x A to A . Thus ϕ α ( x α (0 , 0 , t )) = k · ν ( t ) for all t ∈ K ∗ . The pro of in case s B and F is vir tually the same as the pro o f in ca se G ; w e leav e the details to the rea der. ✷ By 7.1, there exists a θ -inv ar ia nt v aluation ν 1 such that ϕ α satisﬁes 7.2 or 7.3 with ν 1 in place of ν . By 8.10, it fo llows that ν 1 is eq uiv alent to ν . Th us 7.1 1.ii holds. This concludes the pro o f of 7.11. * * * Here, ﬁnally , is a pre c is e version of the remark Tits made in [1 2] that was dis- cussed in the in tro duction. Theorem 8.11 . Bruhat-Tits sp ac es who se buildi ng at inﬁnity (in t he sen s e of 4.14 if the r ank of ∆ is one) is a given Suzuki-R e e building ∆ (in the sen se of 6.8) deﬁne d over a triple ( K , θ , L ) (in c ase B ) or p air ( K , θ ) (in c ases F or G ) ar e classiﬁe d by θ - invariant valuations of K . Equivalent θ -invariant valuations of K c orr esp ond t o e quivalent Bruhat-Tits sp ac es (in the sen s e of 4.5). Pr o of. Let ∆ b e a Suzuki-Ree building deﬁned ov er the triple ( K , θ , L ) in case B or the pair ( K , θ ) in ca ses F or G , and let α and x α be as in 6.1 2 (with resp ect to so me apartment Σ of ∆). Suppo se that ν is a θ -inv aria nt v a luation of K . By 7.4, there exists a v a luation ϕ of the ro ot datum of ∆ ba sed at Σ satisfying 7.2 or 7.3. By 3.16, ϕ is unique up to equip ollence. By 4 .1 6, ϕ determines a unique Br uhat-Tits space ( X , A ) having ∆ as its building at inﬁnity . By 4 .15, a ny v a luation equip ollent to ϕ determines the same Bruhat-Tits space. Suppo se, conversely , that ( X , A ) is a B ruhat-Tits space whose building at inﬁnity is ∆. By 4 .16, ( X , A ) deter mines a v aluation of the ro o t datum of ∆ ba sed at Σ which is unique up to equip ollence. B y 7.1, this equip ollence class deter mines a unique θ -inv aria nt v aluation ν of K such that 7.2 o r 7.3 holds for so me v aluatio n in this equip ollence class. ✷ NON-DISCRETE EUCLIDEAN BUILDINGS 31 9. Appendix The inequalities 7.5 and 7.6 hold by 7.4 and the condition (V1). In this section we g ive an elementary pr o of o f these inequalities w hich might b e of indep endent int eres t. In fact, w e only give a pro of of 7.6; the interested rea der will hav e no trouble a pplying the same strategy to the inequality 7.5. The pro o f we g ive is based on a suggestion of Theo Gr undh¨ ofer. W e supp ose that we are in case G and that ν is a θ -inv ariant v aluation of K . As in 6.1 0, we have (9.1) N ( r, s, t ) = r θ +1 s θ − r t θ − r θ +3 s − r 2 s 2 + s θ +1 + t 2 − r 2 θ +4 for all ( r , s, t ) in the group T . Lemma 9.2. L et ( r, s, t ) ∈ T and su pp ose that t he m inimu m of ν ( r ) , ν ( s ) and ν ( t ) is 0 . Then ν  N ( r, s, t )  = 0 . Pr o of. The Tits endomor phism θ induces a Tits endo mo rphism of ¯ K . W e let ¯ θ denote this endomorphism and let ¯ N b e the map obtained by applying the for m ula 9.1 to the pair ( ¯ K , ¯ θ ) rather than ( K, θ ). By 6.11, ¯ N is anisotropic. ✷ Lemma 9. 3. L et ( r, s, t ) ∈ T , let A = (2 √ 3 + 4 ) ν ( r ) , let B = ( √ 3 + 1) ν ( s ) , let C = 2 ν ( t ) and let M b e t he minimum of A , B and C . Then ν  N ( r, s, t )  = M . Pr o of. W e can as sume that ( r , s, t ) 6 = (0 , 0 , 0). Supp os e ﬁrs t tha t M = A . Then N (1 , s/r θ +1 , t/ r θ +2 ) = N ( r , s, t ) /r 2 θ +4 by 9.1. Mo reov er, ν ( s/r θ +1 ) and ν ( t/r θ +2 ) are b oth non-negative since B ≥ A a nd C ≥ A . Hence ν  N (1 , s/r θ +1 , t/ r θ +2 )  = 0 by 9 .2. It then follows that ν  N ( r, s, t )  = ν ( r 2 θ +4 ) = A = M . Suppo se next that M = C . In this cas e , we obser ve that N ( r /t 2 − θ , s/ t θ − 1 , 1) = N ( r , s, t ) /t 2 . Moreov er, ν ( r /t 2 − θ ) and ν ( s/t θ − 1 ) are b o th non-negative. Hence ν  N ( r /t 2 − θ , s/ t θ − 1 , 1)  = 0 by 9 .2. It then follows that ν  N ( r, s, t )  = ν ( t 2 ) = C = M . It suﬃces now to assume that M = B a nd that B is strictly less than b oth A and C . In this case, ν ( s θ +1 ) is stric tly less than the v alue under ν of each of the remaining six ter ms on the right hand side o f 9 .1. Therefor e ν  N ( r, s, t )  = B = M . ✷ Now let ( r , s, t ) , ( w , u , v ) ∈ T and let M be the s maller of the tw o constants obtained by a pply ing 9.3 ﬁrst to ( r , s, t ) and then to ( w, u, v ). Thus (9.4) ν ( r ) , ν ( w ) ≥ M / (2 √ 3 + 4) as w e ll as (9.5) ν ( s ) , ν ( u ) ≥ M / ( √ 3 + 1) and (9.6) ν ( t ) , ν ( v ) ≥ M / 2 . 32 PETRA HITZELBERGER, LINUS KRAMER AND RICHARD WEISS As in 6.10, we ha ve ( r , s, t ) · ( w , u, v ) = ( r + w , s + u + r θ w, t + v − r u + sw − r θ +1 w ) . Let a = (2 √ 3 + 4) ν ( r + w ), le t b = ( √ 3 + 1) ν ( s + u + r θ w ) and let c = 2 ν ( t + v − r u + sw − r θ +1 w ) . By 9.4 – 9.6, w e hav e (2 √ 3 + 4) ν ( x ) ≥ M for x = r a nd x = w ; ( √ 3 + 1) ν ( x ) ≥ M for x = s , x = u and x = r θ w ; and 2 ν ( x ) ≥ M for x equal to ea ch o f the ﬁve terms in the sum t + v − ru + sw − r θ +1 w. Hence a, b, c ≥ M . B y 9.3, therefore, ν  ( r , s, t ) · ( w , u, v )  ≥ M = min  ν  N ( r, s, t )  , ν  N ( w , u, v )  . References [1] A . Ber enstein and M . Kapovic h, Aﬃne buil dings f or dihedral groups, [2] F. Bruhat and J. Tits, Groupes r ´ eductifs sur un corps l o cal, I. Donn´ ees radicielles v al u ´ ees, Publ. Math. I. H. E. S. 41 (1972), 5-252. [3] T. De M edts, F. Haot, R. Knop and H. V an Maldeghem, On the uniqueness of the unip oten t subgroups of some M oufang sets, in Finite Ge ometries, Gr oups, and Computation (Pingree Pa rk, 2004), pp. 43-66, W alter de Gruyter, Berli n, New Y ork, 2006. [4] B. Kleiner and B. Leeb, Rigidi ty of quasi-i sometries for symm etric spaces and Euclidean buildings, Publ. Math. I. H . E. S. 86 (1997), 115-197. [5] S. Lang, Br uhat-Tits-R¨ aume, Elem. Math. 54 (1999), 45-63. [6] A . Parreau, Immeubles aﬃnes: construction par les normes et ´ etude des isom´ etries, in Crys- tal lo gr aphic Gr oups and their Genera lizations (Kortrijk, 1999), pp. 263-302, Contemp. Math. 262 , Amer. M ath. Soc., Providence , 2000. [7] M . Ronan, Le ct ur es on Buildings , Academic Press, San Diego, 1989. [8] G. Rousseau, Euclidean buil dings, i n S ´ eminair es et Congr` es 18 , Soc. Math. F rance, Pa ris , to appear (http:// hal.archiv es-ouverte s.f r /hal-00094363 ). [9] J. Tits, Les groupes si mples de Suzuki et de Ree, S´ e minair e Bourb aki 6 , Ann ´ ee 1960-61, pp. 65-82, reprinted by So c. Math. F rance, Paris, 1995. [10] J. Tits, Buildings of Spheric al T yp e and Finite BN-Pairs , Springer Lecture Notes in Math. 386 , 1974. [11] J. Tits, Moufang octagons and Ree groups of type 2 F 4 , Amer. J. Math. 105 (1983), 539-594 [12] J. Tits, Immeubles de type aﬃne, in Buildings and t he Ge ometry of D i agr ams (Como, 1984), pp. 159-190, Lecture Notes in Mathematics 118 1 , Spri nger, 1986. [13] J. Tits and R. W eiss, Moufang Polygons , Springer Monographs in Math., Springer, Berlin, New Y ork, Heidelb erg, 2002. [14] R. W eiss, The Structur e of Spheric al Buildings , Princeton Universit y Pr ess, Princeton, 2003. [15] R. W eiss, The Structur e of Aﬃne Buildings , Annals of Math. Studies 168 , Princeton U ni- v ersity Press, Pr inceton, 2009.

Non-discrete Euclidean Buildings for the Ree and Suzuki groups

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