2-dimensional Coxeter groups are biautomatic
22-DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC
ZACHARY MUNRO, DAMIAN OSAJDA † , AND PIOTR PRZYTYCKI ‡ Abstract.
Let W be a -dimensional Coxeter group, that is, a one with m st + m sr + m tr ≤ for all triples of distinct s, t, r ∈ S . We prove that W isbiautomatic. We do it by showing that a natural geodesic language is regular(for arbitrary W ), and satisfies the fellow traveller property. As a consequence,by the work of Jacek Świątkowski, groups acting properly and cocompactlyon buildings of type W are also biautomatic. We also show that the fellowtraveller property for the natural language fails for W = (cid:101) A . Introduction A Coxeter group W is a group generated by a finite set S subject only to relations s = 1 for s ∈ S and ( st ) m st = 1 for m st = m ts ∈ { , , . . . , ∞} . Here the conventionis that m st = ∞ means that we do not impose a relation between s and t . We saythat W is -dimensional if for any triple of distinct elements s, t, r ∈ S , the group (cid:104) s, t, r (cid:105) is infinite. In other words, m st + m sr + m tr ≤ .Consider an arbitrary group G with a finite symmetric generating set S . For g ∈ G , let (cid:96) ( g ) denote the word length of g , that is, the minimal number n suchthat g = s · · · s n with s i ∈ S for i = 1 , . . . , n . Let S ∗ denote the set of all wordsover S . If v ∈ S ∗ is a word of length n , then by v ( i ) we denote the prefix of v oflength i for i = 1 , . . . , n − , and the word v itself for i ≥ n . For ≤ i ≤ j ≤ n by v ( i, j ) we denote the subword of v ( j ) obtained by removing v ( i − . For a word v ∈ S ∗ , by (cid:96) ( v ) we denote the word length of the group element that v represents.We say that G is biautomatic if there exists a regular language L ⊂ S ∗ and aconstant C > satisfying the following conditions (see [ECH +
92, Lem 2.5.5]).(i) For each g ∈ G , there is a word in L representing g .(ii) For each s ∈ S and g, g (cid:48) ∈ G with g (cid:48) = gs, and each v, v (cid:48) ∈ L representing g, g (cid:48) , for all i ≥ we have (cid:96) ( v ( i ) − v (cid:48) ( i )) ≤ C .(iii) For each s ∈ S and g, g (cid:48) ∈ G with g (cid:48) = sg, and each v, v (cid:48) ∈ L representing g, g (cid:48) , for all i ≥ we have (cid:96) ( v ( i ) − s − v (cid:48) ( i )) ≤ C .Our paper concerns the two following well-known open questions (see e.g. [FHT11, § Question 1.
Are Coxeter groups biautomatic?
Question 2.
Are groups acting properly and cocompactly on -dimensional CAT(0) complexes biautomatic?
All Coxeter groups are known to be automatic (i.e. having a regular languagesatisfying (i) and (ii)) by [BH93]. Biautomaticity has been established only inspecial cases: [ECH +
92] (Euclidean and hyperbolic), [NR03] (right-angled), [Bah06]and [CM05] (no Euclidean reflection triangles), [Cap09] (relatively hyperbolic).Question 2 is widely open, while by a recent result of Leary–Minasyan [LM19] itis known that the assumption of -dimensionality is essential. Even in the case of †‡ Partially supported by (Polish) Narodowe Centrum Nauki, UMO-2018/30/M/ST1/00668. ‡ Partially supported by NSERC and AMS. a r X i v : . [ m a t h . G R ] J un Z. MUNRO, D. OSAJDA, AND P. PRZYTYCKI -dimensional buildings, except right-angled and hyperbolic cases, the answer wasknown only in particular instances, e.g. for many (but not all) proper cocompactactions on Euclidean buildings by [GS90], [GS91], [CS95], [Nos00], and [Świ06].To define a convenient language, we need the following. Let W be an arbitraryCoxeter group. For g ∈ W , we denote by T ( g ) ⊆ S the set of s ∈ S satisfying (cid:96) ( gs ) < (cid:96) ( g ) . By [Ron89, Thm 2.16], the group (cid:104) T ( g ) (cid:105) is finite. By w ( g ) wedenote the longest element in (cid:104) T ( g ) (cid:105) (which is unique by [Ron89, Thm 2.15(iii)],and consequently it is an involution). Let Π( g ) = gw ( g ) . By [Ron89, Thm 2.16],we have (cid:96) (Π( g )) + (cid:96) ( w ( g )) = (cid:96) ( g ) .We define the standard language L ⊂ S ∗ for W inductively in the following way.Let v ∈ S ∗ be a word of length n . If v is the empty word, then v ∈ L . Otherwise,let g ∈ W be the group element represented by v and let k = (cid:96) ( w ( g )) . We declare v ∈ L if and only if v ( n − k ) ∈ L and v ( n − k + 1 , n ) represents w ( g ) . In particular, v ( n − k ) represents Π( g ) . It follows inductively that n = (cid:96) ( g ) . Such a language iscalled geodesic . Note that the standard language satisfies part (i) of the definitionof biautomaticity.The paths in W formed by the words in the standard language generalise thenormal cube paths for CAT(0) cube complexes [NR98, §
3] used to prove biau-tomaticity for right-angled (or, more generally, cocompactly cubulated) Coxetergroups [NR03]. Our main result is the following.
Theorem 1.1. If W is a -dimensional Coxeter group, then it is biautomatic with L the standard language. Since the standard language is geodesic and preserved by the symmetries of S preserving the presentation, by [Świ06, Thm 6.7] we have the following immediateconsequence. Corollary 1.2.
Let G be a group acting properly and cocompactly on a building oftype W, where W is a -dimensional Coxeter group. Then G is biautomatic. One element of our proof of Theorem 1.1 is:
Theorem 1.3.
Let W be a Coxeter group. Then its standard language is regular. In other words, the regularity and part (i) of the definition of biautomaticityare satisfied for any Coxeter group W . However, it is not so with part (ii). The (cid:101) A Euclidean group is the Coxeter group with S = { p, r, s, t } , m pr = m rs = m st = m tp = 3 , m ps = m rt = 2 . Theorem 1.4. If W is the (cid:101) A Euclidean group, then its standard language doesnot satisfy part (ii) in the definition of biautomaticity.
Note, however, that by [ECH +
92, Cor 4.2.4], all Euclidean groups, in particu-lar (cid:101) A , are biautomatic (with a different language). Organisation.
In Section 2 we review the basic properties of Coxeter groups. InSection 3 we prove Theorem 1.3. For -dimensional W , we verify parts (iii) and (ii)of the definition of biautomaticity in Sections 4 and 5. This completes the proof ofTheorem 1.1. We finish with the proof of Theorem 1.4 in Section 6.2. Preliminaries By X we denote the Cayley graph of W , that is, the graph with vertex set W and with edges joining each g ∈ W with gs , for s ∈ S . We call such an edge an s -edge . We call gs the s -neighbour of g .For r ∈ W a conjugate of an element of S , the wall W r of r is the fixed pointset of r in X . We call r the reflection in W r (for fixed W r such r is unique). If a -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 3 midpoint of an edge e belongs to a wall W , then we say that W is dual to e (forfixed e such a wall is unique). We say that g ∈ W is adjacent to a wall W , if W is dual to an edge incident to g . Each wall W separates X into two components,and a geodesic edge-path in X intersects W at most once [Ron89, Lem 2.5].For T ⊆ S , each coset g (cid:104) T (cid:105) ⊆ X for g ∈ W is a T -residue . A geodesic edge-pathin X with endpoints in a residue R has all its vertices in R [Ron89, Lem 2.10].We say that a wall W intersects a residue R if W separates some elements of R .Equivalently, W is dual to an edge with both endpoints in R . Theorem 2.1 ([Ron89, Thm 2.9]) . Let W be a Coxeter group. Any residue R of X contains a unique element h with minimal (cid:96) ( h ) . Moreover, for any g ∈ R wehave (cid:96) ( h ) + (cid:96) ( h − g ) = (cid:96) ( g ) . As introduced in Section 1, for g ∈ W we denote by T ( g ) ⊆ S the set of s ∈ S satisfying (cid:96) ( gs ) < (cid:96) ( g ) . Let R be the T ( g ) -residue containing g . By [Ron89,Thm 2.16], the group (cid:104) T ( g ) (cid:105) is finite and, for w ( g ) the longest element in (cid:104) T ( g ) (cid:105) ,the unique element h ∈ R from Theorem 2.1 is Π( g ) = gw ( g ) . In particular, wehave (cid:96) (Π( g )) + (cid:96) ( w ( g )) = (cid:96) ( g ) . Consequently, if W is -dimensional, then for each g ∈ W we have | T ( g ) | = 1 or .For g ∈ W , let W ( g ) be the set of walls W in X that separate g from theidentity element id ∈ W and such that there is no wall W (cid:48) separating g from W .By the following Parallel Wall Theorem, there exists a bound on the distance in X between g and each of the walls of W ( g ) . Theorem 2.2 ([BH93, Thm 2.8]) . Let W be a Coxeter group. There is a constant Q = Q ( W ) such that for any g ∈ W and a wall W at distance > Q from g in X ,there is a wall W (cid:48) separating g from W . By X we denote the Cayley complex of W . It is the piecewise Euclidean -complex with -skeleton X , all edges of length , and a regular m st -gon spannedon each { s, t } -residue with m st < ∞ . If W is -dimensional, then X is CAT(0) (seethe link condition in [BH99, § II.5.24]). Walls in X extend to (convex) walls in X, which still separate X . 3. Regularity A finite state automaton over S (FSA) is a directed graph Γ with vertex set V ,edge set E ⊆ V × V , an edge labeling φ : E → P ( S ∗ ) (the power set of S ∗ ), adistinguished set of start states S ⊆ V , and a distinguished set of accept states F ⊆ V . A word v ∈ S ∗ is accepted by Γ if there exists a decomposition v = v · · · v m of v into subwords and an edge-path e · · · e m in Γ such that e has initial vertexin S , e m has terminal vertex in F , and v i ∈ φ ( e i ) for each i = 0 , . . . , m . A subsetof S ∗ is a regular language if it is the set of accepted words for some FSA over S .The proof of regularity of the standard language relies on Theorem 2.2 and thefollowing lemma. Lemma 3.1.
Let W be a Coxeter group. Let g ∈ W , let T ⊆ S be such that (cid:104) T (cid:105) isfinite, and let w be the longest element in (cid:104) T (cid:105) . Then T ( gw ) = T if and only if(i) T is disjoint from T ( g ) , and(ii) for each t ∈ S \ T , the wall dual to ( gw, gwt ) does not lie in W ( g ) . Note that condition (i) could be written equivalently as: for each t ∈ T , the walldual to ( g, gt ) does not lie in W ( g ) . Proof of Lemma 3.1.
Suppose first T ( gw ) = T . Then, for R the T -residue con-taining gw , by the discussion after Theorem 2.1, the unique element h ∈ R withminimal (cid:96) ( h ) is g . Thus for each t ∈ T we have (cid:96) ( gt ) > (cid:96) ( g ) and so condition (i) Z. MUNRO, D. OSAJDA, AND P. PRZYTYCKI holds. Furthermore, for t ∈ S \ T , the wall W dual to the edge ( gw, gwt ) does notseparate gw from id . Additionally, the wall W cannot separate gw from g : If itdid, then after conjugating by ( gw ) − , the reflection in W could become simulta-neously the generator t and a word in the elements of T , contradicting t ∈ S \ T by [Ron89, Lem 2.1(ii)]. Thus W does not separate g from id , and so condition (ii)holds.Conversely, suppose that T ⊆ S has finite (cid:104) T (cid:105) and satisfies conditions (i) and (ii).Then, by condition (i), for R the T -residue containing g , we have that the minimalword length element h ∈ R from Theorem 2.1 coincides with g , and so the maximalword length element is gw . Consequently, we have T ( gw ) ⊇ T . Suppose, forcontradiction, that there is t ∈ T ( gw ) \ T . Then the wall W dual to the edge ( gw, gwt ) separates gw from id . Following the argument in the previous paragraph, W does not separate gw from g , so it separates g from id . Furthermore, if a wall W (cid:48) separated W from g , then W (cid:48) would also have to separate gw from g , contradicting (cid:96) ( g )+ (cid:96) ( w ) = (cid:96) ( gw ) . Consequently, W ∈ W ( g ) , which contradicts condition (ii). (cid:3) We now define an FSA Γ over S that will accept exactly the standard language. Definition 3.2.
Let Q be the constant from Theorem 2.2. For g ∈ W , let M Q ( g ) be the set of walls in X intersecting the closed ball in X of radius Q centred at g .By Theorem 2.2, we have W ( g ) ⊆ M Q ( g ) .Consider the set ˆ V of pairs of the form ( g, c ) , where g ∈ W , and c is any functionfrom M Q ( g ) to { , } . For g, h ∈ W, there exists a Cayley graph automorphism ϕ gh : X → X taking g to h, given by the left-multiplication by hg − . This mapinduces a bijection ϕ ∗ gh : M Q ( g ) → M Q ( h ) . We define an equivalence relation ∼ on ˆ V by ( g, c ) ∼ ( h, c (cid:48) ) if c = c (cid:48) ◦ ϕ ∗ gh . We take the vertices of our FSA Γ to be V = ˆ V / ∼ .In any equivalence class of ∼ , there is exactly one representative of the form (id , c ) . Suppose that we have T ⊆ S such that (cid:104) T (cid:105) is finite. Let w be the longestelement of (cid:104) T (cid:105) . If(i) for each t ∈ T, the wall dual to (id , t ) lies outside c − (1) , and(ii) for each t ∈ S \ T, the wall dual to ( w, wt ) lies outside c − (1) ,then we put an edge e in Γ from [(id , c )] to [( w, c (cid:48) )] , where c (cid:48) is defined by being on all walls in M Q ( w ) that(a) lie in c − (1) or intersect the residue (cid:104) T (cid:105) , and(b) are not separated from w by a wall satisfying (a).We let the label φ ( e ) to be the set of all minimal length words representing w .We let all states be accept states of Γ and let the set of start states S containjust [(id , c )] , where c is the constant function. Proof of Theorem 1.3.
Let Γ be the FSA from Definition 3.2, and let L be thestandard language. We argue inductively on j ≥ that, among the words v ∈ S ∗ of length ≤ j , • Γ accepts exactly the words in L , and • the accept state of each such word v is [( g, c )] , where v represents g and c takes value exactly on the walls in W ( g ) .This is true for j = 0 by our choice of S . Now let n > and suppose that we haveverified the inductive hypothesis for all j < n . Let v be a word in S ∗ of length n .Suppose first that v is a word in L representing g ∈ W . By the definition of L ,for k = (cid:96) ( w ( g )) , we have v ( n − k ) ∈ L . Moreover, v ( n − k + 1 , n ) represents w ( g ) .By the inductive hypothesis, Γ accepts v ( n − k ) . Furthermore, v ( n − k ) labels someedge-path in Γ from S to [(Π( g ) , c )] , where c takes value exactly on the walls in -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 5 W (Π( g )) . Let T = T ( g ) . By Lemma 3.1, applied replacing g with Π( g ) , we havethat(i) for each t ∈ T , the wall dual to (Π( g ) , Π( g ) t ) does not lie in W (Π( g )) , and(ii) for each t ∈ S \ T, the wall dual to ( g, gt ) does not lie in W (Π( g )) .Thus, Γ has an edge from [(Π( g ) , c )] to [( g, c (cid:48) )] , labelled by v ( n − k + 1 , n ) , and so Γ accepts v . Furthermore, since W ( g ) ⊆ M Q ( g ) , and by conditions (a) and (b) inDefinition 3.2, c (cid:48) takes value exactly on the walls in W ( g ) .Conversely, let v be accepted by Γ and suppose that v = v · · · v m as in thedefinition of an accepted word. By the inductive hypothesis, the word v · · · v m − belongs to L and represents g ∈ W such that e m starts at [( g, c )] , where c takesvalue exactly on the walls in W ( g ) . By definition of the edges, v m represents thelongest element w in some finite (cid:104) T (cid:105) , and g and T satisfy conditions (i) and (ii)in Definition 3.2. Thus, g and T satisfy conditions (i) and (ii) of Lemma 3.1.Consequently, we have T = T ( gw ) , and so v belongs to L . (cid:3) g and sg Lemma 4.1.
Let W be a -dimensional Coxeter group. Then its standard languagesatisfies part (iii) of the definition of biautomaticity. We will need the following.
Sublemma 4.2.
Let W be a -dimensional Coxeter group. There is a constant D = D ( W ) such that for any wall W adjacent to id , any f ∈ W adjacent to W , andany vertices h, h (cid:48) ∈ W on geodesic edge-paths from id to f satisfying (cid:96) ( h ) = (cid:96) ( h (cid:48) ) ,we have (cid:96) ( h − h (cid:48) ) < D .Proof. Let Q = Q ( W ) be the constant from Theorem 2.2. Suppose that h, h (cid:48) ∈ W lie on geodesic edge-paths γ, γ (cid:48) from id to f and satisfy (cid:96) ( h ) = (cid:96) ( h (cid:48) ) . Then eachvertex g ∈ W of γ lies at distance ≤ Q from W in X , since otherwise there wouldbe a wall W (cid:48) separating g from W , and so W (cid:48) would intersect γ at least twice.Since W is -dimensional, we have that X is a CAT(0) space and the extensionof W to X (for which we keep the same notation) is a convex tree. Let x, y ∈W be the midpoints of the edges dual to W incident to id , f , respectively. Let N ( W ) be the closed Q -neighbourhood of W in X , w.r.t. the CAT(0) metric. Notethat N ( W ) is quasi-isometric to W , so in particular N ( W ) is Gromov-hyperbolic.Moreover, since X and X are quasi-isometric, we have that γ ⊂ N ( W ) is a ( λ, (cid:15) ) -quasigeodesic, where the constants λ, (cid:15) depend only on W . Consequently, by thestability of quasi-geodesics, for a constant C = C ( W ) , there is a point z on thegeodesic xy with | h, z | ≤ C . Analogously, there is a vertex h (cid:48)(cid:48) on γ (cid:48) with | z, h (cid:48)(cid:48) | ≤ C ,and so | h, h (cid:48)(cid:48) | ≤ C .Thus, since X and X are quasi-isometric, there is a constant D = D ( W ) with (cid:96) ( h − h (cid:48)(cid:48) ) < D . By the triangle inequality in X , we have | (cid:96) ( h ) − (cid:96) ( h (cid:48)(cid:48) ) | < D .Thus by (cid:96) ( h ) = (cid:96) ( h (cid:48) ) we have (cid:96) ( h (cid:48)(cid:48)− h (cid:48) ) < D . Consequently, (cid:96) ( h − h (cid:48) ) < D , asdesired. (cid:3) Proof of Lemma 4.1.
Let L be the standard language. Let D be the constant fromSublemma 4.2. Let K be the maximal word length of the longest element of a finite (cid:104) T (cid:105) over all T ⊆ S , and let C = max { K, D } .We prove part (iii) of the definition of biautomaticity inductively on (cid:96) ( g ) , wherewe assume without loss of generality (cid:96) ( sg ) > (cid:96) ( g ) . If g = id , then there is nothingto prove. Suppose now g (cid:54) = id , and let W be the wall in X dual to the s -edgeincident to id .Assume first that g is not adjacent to W . Let W (cid:48) be a wall adjacent to g separating g from id . Then W (cid:48) also separates g from s . Consequently, s W (cid:48) separates Z. MUNRO, D. OSAJDA, AND P. PRZYTYCKI sg from id . Conversely, if a wall W (cid:48) is adjacent to sg and separates sg from id , thenit also separates sg from s , and so s W (cid:48) separates g from id . Consequently, T ( sg ) = T ( g ) and so w ( g ) = w ( sg ) , hence Π( sg ) = s Π( g ) . In other words, for v, v (cid:48) ∈ L representing g, sg, and k = (cid:96) ( w ( g )) , the words v (cid:48) ( (cid:96) ( sg ) − k ) and sv ( (cid:96) ( g ) − k ) representthe same element s Π( g ) of W . Then part (iii) of the definition of biautomaticityfor g follows inductively from part (iii) for Π( g ) , for i < (cid:96) ( sg ) − k , or from thedefinition of K , for i ≥ (cid:96) ( sg ) − k .Secondly, assume that g is adjacent to W . Then ( g, sg ) is an edge of X . Let f = sg and for ≤ i ≤ (cid:96) ( g ) let h, h (cid:48) be the elements of W represented by sv ( i ) and v (cid:48) ( i + 1) . Then, by the definition of D , we have (cid:96) ( v ( i ) − sv (cid:48) ( i + 1)) < D , asdesired. (cid:3) g and gs For g ∈ W and k ≥ , we set Π k ( g ) = k (cid:122) (cid:125)(cid:124) (cid:123) Π ◦ · · · ◦ Π( g ) . The main result of thissection is the following. Proposition 5.1.
Let W be a -dimensional Coxeter group. Let g, g (cid:48) ∈ W be suchthat g (cid:48) ∈ g (cid:104) s, t (cid:105) for some s, t ∈ S with m st < ∞ (possibly s = t ). Then there are ≤ k, k (cid:48) ≤ with k + k (cid:48) > , such that Π k (cid:48) ( g (cid:48) ) ∈ Π k ( g ) (cid:104) p, r (cid:105) for some p, r ∈ S with m pr < ∞ (possibly p = r ). We obtain the following consequence, which together with Theorem 1.3 andLemma 4.1 completes the proof of Theorem 1.1.
Corollary 5.2.
Let W be a -dimensional Coxeter group. Then its standard lan-guage satisfies part (ii) of the definition of biautomaticity.Proof. As before, let K be the maximal word length of the longest element of afinite (cid:104) T (cid:105) over all T ⊆ S . Assume without loss of generality (cid:96) ( gs ) > (cid:96) ( g ) .Let ≤ i ≤ (cid:96) ( g ) . By Proposition 5.1, there is ≤ j ≤ (cid:96) ( g ) with | j − i | ≤ K and ≤ i (cid:48) ≤ (cid:96) ( g ) + 1 such that v ( j ) and v (cid:48) ( i (cid:48) ) represent elements of W in a commonfinite residue. Consequently, we have (cid:96) (cid:0) v ( j ) − v (cid:48) ( i (cid:48) ) (cid:1) ≤ K, and so in particular | j − i (cid:48) | ≤ K . Therefore (cid:96) (cid:0) v ( i ) − v (cid:48) ( i ) (cid:1) ≤ | i − j | + (cid:96) (cid:0) v ( j ) − v (cid:48) ( i (cid:48) ) (cid:1) + | i (cid:48) − i | ≤ K . (cid:3) In the proof of Proposition 5.1 we will use the following truncated piecewise Eu-clidean structure on the Cayley complex X of W . Consider the function q : { , , . . . , ∞} →{ , , . . . , ∞} , defined as q ( m ) = , for m = 4 , , , for m ≥ ,m, otherwise.In each regular m -gon of X in the usual piecewise Euclidean structure (each such m -gon corresponds to an { s, t } -residue with m st = m ), we replace each triangleof its barycentric subdivision with the triangle of the barycentric subdivision ofa regular q ( m ) -gon. For example, we replace each triangle of the barycentricsubdivision of a regular -gon with the one of an -gon, i.e. of angles π , π , π . Lemma 5.3.
The truncated piecewise Euclidean structure satisfies the link condi-tion , i.e. each loop in the link of a vertex has length ≥ π . Here we declare each edge of the link to have the length equal to the angle of thetriangle that it corresponds to. Similarly, for two edges e, e (cid:48) incident to a vertex v ,by their angle at v we mean the distance in the link of v between the vertices that e, e (cid:48) correspond to. The advantage of the truncated structure over the usual one isthat the angles between walls increase. -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 7 Proof of Lemma 5.3.
When we pass from the usual piecewise Euclidean structureon the barycentric subdivision of X to the truncated one, the angles at the barycen-tres of edges all stay equal to π . Furthermore, the angles at the barycentres of -cells increase. It remains to consider a vertex v ∈ X . After merging pairs ofedges coming from the same polygon of X , the link L of v has a vertex for each s ∈ S and an edge of length (cid:0) − q ( m st ) (cid:1) π ≥ π for each s, t ∈ S with m st < ∞ . Inparticular, all the loops in L of combinatorial length ≥ have metric length ≥ π .To obtain the same for loops in L of combinatorial length , we need to verify thatfor each triple of distinct s, t, r ∈ S , we have( ∗ ) q ( m st ) + 1 q ( m tr ) + 1 q ( m sr ) ≤ . If q ( m st ) , q ( m tr ) , q ( m sr ) (cid:54) = 2 , then ( ∗ ) holds. If q ( m st ) , q ( m tr ) (cid:54) = 2 and q ( m sr ) =2 , then m st , m tr (cid:54) = 2 and m sr = 2 . Since W is -dimensional, we have m st , m tr ≥ or m st ≥ or m tr ≥ . We then have, respectively, q ( m st ) , q ( m tr ) ≥ or q ( m st ) ≥ or q ( m tr ) ≥ , and so ( ∗ ) holds in this case as well. Finally, if q ( m st ) = q ( m sr ) = 2 ,then m st = m sr = 2 , contradicting the -dimensionality of W . (cid:3) Lemma 5.4.
Let W be a -dimensional Coxeter group. Let γ, γ (cid:48) be geodesic edge-paths in X with common endpoints. Suppose that there are walls W i in X with i = 1 , , , such that γ intersects them in the opposite order to γ (cid:48) , and that W is the middle one in both of these orders. For i = 1 , , let θ i be the angle in thetruncated structure at x i = W ∩ W i formed by the segments in W , W i from x i to γ ∩ W and γ ∩ W i . Then θ + θ < π . γ γ (cid:48) W W W θ θ x x Figure 1.
Lemma 5.4See Figure 1 for an illustration. Note that in the definition of either θ i we couldreplace γ by γ (cid:48) . Proof.
Let D be a reduced diagram in X with boundary γ − γ (cid:48) (see for example[LS77, § V.1–2]). By Lemma 5.3, the diagram D with the path metric inducedfrom the truncated Euclidean structure on X is a CAT(0) space. The diagram D contains a geodesic triangle formed by the segments of the walls W i joining theirthree intersection points. Its angles indicated in Figure 1 equal θ , θ . Since theseangles do not exceed the angles of the comparison triangle in the Euclidean plane[BH99, II.1.7(4)], we have θ + θ < π . (cid:3) Z. MUNRO, D. OSAJDA, AND P. PRZYTYCKI
Corollary 5.5.
Let W be a -dimensional Coxeter group. Let f ∈ W with T ( f ) = { s, t } , with s (cid:54) = t . Let h = Π( f ) and let R be the { s, t } -residue containing f and h .Let g ∈ R and let m be the distance in X between g and h . Suppose T ( g ) = { s, r } with r (cid:54) = s, t . Then:(i) m ≤ .(ii) If m = 3 , then m sr = 2 .(iii) If m st = 3 and m = 2 , then m sr = 2 .(iv) If m st = 4 , then m ≤ .(v) If m = m st − , then m st ≤ , and for m st = 3 we have m sr = 2 .Proof. Note that T ( g ) = { s, r } implies in particular g (cid:54) = f, h . Let γ be thegeodesic edge-path in X from f to h not containing g . Let γ be the length m sr geodesic edge-path with vertices in the { s, r } -residue containing g , starting at g with the r -edge. Let γ be any geodesic edge-path from f to id containing γ . Let γ (cid:48) be any geodesic edge-path from f to id containing γ . Let W be the first wallintersecting γ . Let W be the wall dual to the s -edge incident to g . Let W bethe wall dual to the r -edge incident to g . See Figure 2. Note that W does notintersect R (since then W and W would intersect twice in X ) and, analogously, W does not intersect the { s, r } -residue of g . Consequently, we are in the setup ofLemma 5.4. ss t t fh r g W W W r st γγ (cid:48) id R Figure 2.
Corollary 5.5Observe that we have θ = ( m − πq ( m st ) and θ = ( m sr − πq ( m sr ) .To prove part (i), assume m ≥ . We then have θ ≥ π . However, θ ≥ π , whichcontradicts Lemma 5.4.For part (ii), if m = 3 then we only have θ ≥ π . However, assuming m sr ≥ ,we would have θ ≥ π , which also contradicts Lemma 5.4.For part (iii), if m = 2 and m st = 3 , then we have θ = π . Assuming m sr ≥ ,we would have θ ≥ π as before, which contradicts Lemma 5.4.To prove part (iv), if we had m st = 4 and m = 3 , then θ ≥ π and θ ≥ π wouldalso contradict Lemma 5.4.For part (v), assume m = m st − . The case m st ≥ is excluded by part (i),and the case m st = 4 is excluded by part (iv). For m st = 3 we have m sr = 2 bypart (iii). (cid:3) Proof of Proposition 5.1. If s = t , then without loss of generality s ∈ T ( g ) , and wecan take k = 1 , k (cid:48) = 0 . -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 9 Assume now s (cid:54) = t . Let R = g (cid:104) s, t (cid:105) , and let f, h ∈ R be the elements withmaximal and minimal word length, respectively. Let m, m (cid:48) be the distances in X between h and g, g (cid:48) , respectively. We can assume Π( g ) , Π( g (cid:48) ) / ∈ R . Then in partic-ular m, m (cid:48) (cid:54) = m st and for m (cid:54) = 0 we have | T ( g ) | = 2 and T ( g ) contains exactly oneof s, t . Without loss of generality we suppose then T ( g ) = { s, r } .Note that from Corollary 5.5(i) it follows that m ≤ . Furthermore, by Corol-lary 5.5(ii) if m = 3 , then m sr = 2 . An analogous statement holds for m (cid:48) . Case 1: m = 3 , or m = 2 and m sr ≥ . If m = 3 , then denoting by ˆ g the s -neighbour of g , we have T (ˆ g ) = { t, r } . Since m sr = 2 , we have m tr ≥ . s sst t t fh RR (cid:48) r g ˆ g r r r s (a) (b) s sst t t fh RR (cid:48) r g ˆ g r r r s u ˆ u ˆ u u ˆ h tt tt ss r rr Figure 3.
Proof of Proposition 5.1, Case 1.Applying Corollary 5.5(v), with f replaced by ˆ g and g replaced by the t -neighbourof ˆ g , gives m tr = 3 , and so m st ≥ . Consequently, in X we have the configurationdescribed in Figure 3(a), where the height of the vertices in the figure correspondsto their word length (i.e. their distance from id in X ). Each edge-path in X labelled sr or trt , whose endpoints’ word length differs by or , respectively,shares its endpoints with an edge-path labelled rs or rtr with uniquely determinedword lengths of vertices. Thus the configuration described in Figure 3(a) extendsto the configuration in Figure 3(b). In particular, we have m (cid:48) (cid:54) = 3 , since otherwise m sr ≥ in the analogy to the first paragraph of Case 1. Consequently, m (cid:48) ≤ .Consider any of the two vertices labelled by u in Figure 3(b). Note that T ( u ) = { t } , since having | T ( u ) | = 2 would force the t -neighbour ˆ u of u to have | T (ˆ u ) | ≥ .This implies that Π ( g ) lies on the lower { s, t } -residue R (cid:48) in Figure 3(b). Further-more, note that T ( h ) = { r } , since having T ( h ) = { r, p } for some p ∈ S would forcethe r -neighbour ˆ h of h to have T (ˆ h ) = { t, p } , contradicting Corollary 5.5(v) with g replaced by ˆ h , and f replaced by the s -neighbour of ˆ h . Consequently, in any of thecases m (cid:48) = 0 , , , there is k (cid:48) ≤ with Π k (cid:48) ( g (cid:48) ) ∈ R (cid:48) , as desired.If m = 2 and m sr ≥ , then the same proof goes through with the followingminor changes. Namely, m sr = 3 and m tr = 2 follow from Corollary 5.5(v) appliedwith f replaced by g and g replaced by the s -neighbour of g . The remaining partof the proof is the same, with s and t interchanged, except that it is Π( g ) insteadof Π ( g ) that lies in R (cid:48) . Case 2: m = 2 and m sr = 2 . We have m tr ≥ and the configuration from Figure 4 inside X . Note that if m (cid:48) = 2 , then we can assume T ( g (cid:48) ) = { t } . Indeed, if T ( g (cid:48) ) = { t, p } , then we canassume m tp = 2 since otherwise interchanging g, g (cid:48) we can appeal to Case 1. Thus p (cid:54) = r , and so the t -neighbor ˆ g (cid:48) of g (cid:48) has | T (ˆ g (cid:48) ) | ≥ , which is a contradiction. sss t tt fhg r r r r RR (cid:48) Figure 4.
Proof of Proposition 5.1, Case 2.Consequently both Π( g ) and Π k (cid:48) ( g (cid:48) ) for some k (cid:48) ≤ lie in the { t, r } -residue R (cid:48) from Figure 4. This completes Case 2.Note that if, say, m = 1 , m (cid:48) = 0 , then we can take k = 1 , k (cid:48) = 0 . Thus it remainsto consider the case where m (cid:48) = m = 1 . Case 3: m (cid:48) = m = 1 , and T ( g (cid:48) ) = { t, r } . In other words, the second element of T ( g (cid:48) ) coincides with that of T ( g ) .If one of m sr , m tr , say m sr , equals , then we can take k = 1 , k (cid:48) = 0 , and weare done. If m sr = m tr = 3 , then we can take k = k (cid:48) = 1 . It remains to considerthe case, where, say, m sr ≥ , m tr ≥ . Let γ , γ (cid:48) be the geodesic edge-paths from f to g, g (cid:48) , respectively. If m tr ≥ , then we apply Lemma 5.4 with any γ startingwith γ m sr (cid:122) (cid:125)(cid:124) (cid:123) rsr · · · , and any γ (cid:48) starting with γ (cid:48) m tr (cid:122) (cid:125)(cid:124) (cid:123) rtr · · · . We take W , W , W to be thewalls dual to r -edges incident to g, h, g (cid:48) , respectively. Then θ , θ ≥ π , which is acontradiction. Analogously, if m sr ≥ , then θ ≥ π , θ ≥ π , contradiction.We can thus assume m tr = 3 , and m sr = 4 or . In particular, m st ≥ . Wenow apply Corollary 5.5, with f replaced by the r -neighbour u of h and g replacedby the s -neighbour ˆ u of u , see Figure 5. Since T ( u ) = { s, t } with m st ≥ and T (ˆ u ) = { t, r } with m tr = 3 , Corollary 5.5(v) yields a contradiction. Case 4: m (cid:48) = m = 1 , and T ( g (cid:48) ) = { t, p } for some p (cid:54) = r . If m sr = m tp = 2 , then we can take k (cid:48) = k = 1 and we are done. We now focuson the case m sr ≥ and m tp ≥ . By Corollary 5.5(v), applied with f replaced by g and g replaced by h , we obtain m sr = 3 and m rp = 2 . Let ˆ h be the p -neighbourof h . We then apply Lemma 5.4 to geodesic edge-paths γ, γ (cid:48) from g to id , where γ starts with the length m sr edge-path in the { s, r } -residue of g starting with the r -edge, and γ (cid:48) starts with the s -edge, the p -edge, followed by the length m tr edge-path in the { t, r } residue of ˆ h starting with the t -edge. See Figure 6. We considerthe walls W , W dual to the r -edges incident to g, h , respectively, and W dual tothe t -edge incident to ˆ h . We have θ = π , θ ≥ π , which is a contradiction.It remains to consider the case where, say, m sr ≥ and m tp = 2 . Then againby Corollary 5.5(v), applied with f replaced by g and g replaced by h , we obtain -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 11 sss t t t fh rs sttr u ˆ ug Figure 5.
Proof of Proposition 5.1, Case 3. sss t t t fh r g ˆ h R W W W p pr id r t γ γ (cid:48) Figure 6.
Proof of Proposition 5.1, Case 4, m sr ≥ and m tp ≥ . m sr = 3 and m rp = 2 . Let u be the r -neighbour of h . Then Π( g ) lies in the { p, s } -residue R (cid:48) of u . Let ˆ h = Π( g (cid:48) ) , and let ˆ u be the p -neighbour of u , see Figure 7.We have m sp ≥ and so by Corollary 5.5(v), applied with f replaced by u and g replaced by ˆ u , we obtain T (ˆ u ) = { s } . We claim that T (ˆ h ) = { r } and so Π(ˆ h ) also lies in R (cid:48) , finishing the proof. To justify the claim, suppose T (ˆ h ) = { r, q } with q (cid:54) = r . If m rq ≥ , then we obtain a contradiction as in the previous paragraph. If m rq = 2 , then q (cid:54) = s and so T (ˆ u ) = { s, q } , which is a contradiction. This justifiesthe claim and completes Case 4. (cid:3) sss t t t fh r g ˆ h R u ˆ u R (cid:48) ? pr ps s r Figure 7.
Proof of Proposition 5.1, Case 4, m sr ≥ and m tp = 2 .6. (cid:101) A Euclidean group
In this section it will be convenient to view the Cayley graph X of the (cid:101) A Coxeter group W as the dual graph to its Coxeter complex, which is the followingsubdivision of R . (The reader might find it convenient to relate this subdivisionintro tetrahedra with the standard subdivision of R into unit cubes.) Its verticesare triples of integers ( x, y, z ) that are all odd or all even. Edges connect each vertex ( x, y, z ) to vertices of the form ( x ± , y, z ) , ( x, y ± , z ) , ( x, y, z ± , ( x ± , y ± , z ± .See for example [Mun19, Thm A], where this Coxeter complex is described as asubdivision of the hyperplane x + x + x + x = 0 in R , and the linear isomorphismwith our subdivision of R is given by ( x, y, z ) (cid:55)→ ( x + y + z, x − y − z, y − z − x, z − x − y ) .Tetrahedra are spanned (up to permuting the coordinates) on cliques with ver-tices ( x, y, z − , ( x, y, z + 1) , ( x + 1 , y − , z ) , ( x + 1 , y + 1 , z ) . Each such tetrahe-dron has exactly two edges of length , and the segment e = (( x, y, z ) , ( x + 1 , y, z )) joining their centres has length . We can equivariantly embed X into R bymapping each vertex into the centre of a tetrahedron, and mapping each edgeaffinely. Consequently, we can identify elements g ∈ W with segments of the form e g = (( x, y, z ) , ( x +1 , y, z )) , where y + z is odd, up to permuting the coordinates. Weidentify id ∈ W with e id = ((0 , , , (0 , , . In particular, the point O = (0 , , belongs to the identity tetrahedron. Note that for each g ∈ W, s ∈ S , the segments e g , e gs are incident. Furthermore, walls in X extend to subcomplexes of R iso-metric to Euclidean planes, and such a wall is adjacent to g ∈ W if and only if itcontains a face of the tetrahedron containing e g . Lemma 6.1.
Let | x | + 1 < y < z . Let g ∈ W be such that(i) e g = (( x , y , z ) , ( x + 1 , y , z )) , or(ii) e g = (( x , y , z ) , ( x , y , z + 1)) .Then (cid:96) ( w ( g )) equals, respectively,(i) , or(ii) . -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 13 Furthermore, e Π( g ) is equal to the translate of e g by, respectively,(i) (0 , − , − , or(ii) (0 , , − .Proof. In case (i), suppose first that x + y is even. Then e g lies in the tetrahedronwith vertices ( x , y , z − , ( x , y , z + 1) , ( x + 1 , y − , z ) , ( x + 1 , y + 1 , z ) .The walls adjacent to g are the hyperplanes containing the faces of this tetrahedron,which are x + y = x + y , x − y = x − y , x + z = x + 1 + z , x − z = x + 1 − z .Projecting e g , O, and these walls onto the xy plane (Figure 8(a)), or the xz plane(Figure 8(b)), we obtain that e g is separated from O exactly by the first and fourthamong these walls. O O x − y = x − y x + y = x + y x − z = x + − z x + z = x + + z e g e g (a) (b) sector | x | ≤ y sector | x | ≤ z Figure 8.
Proof of Lemma 6.1, case (i), x + y even. e g e g + ( , − , − ) ( x , y , z ) ( x + 1 , y , z ) x + y = x + y x − z = x + − z π Figure 9.
Proof of Lemma 6.1, case (i), x + y even: the two walls. Consequently, gT ( g ) g − consists of the reflections in the first and fourth of thesewalls. These reflections preserve the cube spanned by e g and its translates by (0 , − , , (0 , , − , and (0 , − , − , see Figure 9. The longest element (of length )in the group that these reflections generate maps e g to its translate by (0 , − , − .Secondly, suppose that x + y is odd. Then e g lies in the tetrahedron withvertices ( x , y − , z ) , ( x , y + 1 , z ) , ( x + 1 , y , z − , ( x + 1 , y , z + 1) . Thusthe walls adjacent to g are x + y = x + 1 + y , x − y = x + 1 − y , x + z = x + z , x − z = x − z . Hence, as illustrated in Figure 10(a,b), e g is separatedfrom O exactly by the second and third among these walls. Consequently, gT ( g ) g − consists of the reflections in the second and third of these walls. The longest element(of length ) in the group they generate maps e g to its translate by (0 , − , − asbefore. O O x − y = x + − y x + y = x + + y x − z = x − z x + z = x + z e g e g (a) (b) sector | x | ≤ y sector | x | ≤ z Figure 10.
Proof of Lemma 6.1, case (i), x + y odd.In case (ii), suppose first that y + z is odd. Then e g lies in the tetrahedron withvertices ( x , y − , z ) , ( x , y + 1 , z ) , ( x − , y , z + 1) , ( x + 1 , y , z + 1) . Thusthe walls adjacent to g are x + z = x + z , x − z = x − z , y + z = y + z +1 , y − z = y − z − . Hence, as illustrated in Figure 11(a,b), e g is separated from O exactlyby the first and second among these walls. Consequently, gT ( g ) g − consists ofthe reflections in the first and second of these walls. These reflections commuteand preserve the square spanned by e g and its translate by (0 , , − . The longestelement in the group these reflections generate (i.e. their composition) maps e g toits translate by (0 , , − .Secondly, suppose that y + z is even. Then e g lies in the tetrahedron withvertices ( x − , y , z ) , ( x + 1 , y , z ) , ( x , y − , z + 1) , ( x , y + 1 , z + 1) . Thusthe walls adjacent to g are x + z = x + z + 1 , x − z = x − z − , y + z = y + z , y − z = y − z . Hence, as illustrated in Figure 12(a,b), e g is separatedfrom O exactly by the third and fourth among these walls. Consequently, gT ( g ) g − consists of the (commuting) reflections in the third and fourth of these walls. Thelongest element in the group they generate maps e g to its translate by (0 , , − asbefore. (cid:3) Proof of Theorem 1.4.
Let L be the standard language. For each C > considerthe following g, g (cid:48) ∈ W with incident segments e g = (( x , y , z ) , ( x + 1 , y , z )) , e g (cid:48) = (( x , y , z ) , ( x , y , z + 1)) with x , z even and y odd, satisfying | x | + C < y ≤ z − C . Suppose that g, g (cid:48) are represented by v, v (cid:48) ∈ L of length N, N (cid:48) (which differ by ). By Lemma 6.1, for -DIMENSIONAL COXETER GROUPS ARE BIAUTOMATIC 15 O O x − z = x − z x + z = x + z y − z = y − z − y + z = y + z + e g e g (a) (b) sector | x | ≤ z sector 0 ≤ y ≤ z Figure 11.
Proof of Lemma 6.1, case (ii), y + z odd. O O x − z = x − z − x + z = x + z + y − z = y − z y + z = y + z e g e g (a) (b) sector | x | ≤ z sector 0 ≤ y ≤ z Figure 12.
Proof of Lemma 6.1, case (ii), y + z even. n, n (cid:48) ≤ C we have that v ( N − n ) represents the element of W corresponding to thesegment e g − n (0 , , and v (cid:48) ( N (cid:48) − n (cid:48) ) represents the element of W corresponding tothe segment e g (cid:48) − n (cid:48) (0 , , . In particular, for i = 3 n = 2 n (cid:48) , we see that the segmentscorresponding to v ( N − i ) and v (cid:48) ( N (cid:48) − i ) are (( x , y − n, z − n ) , ( x +1 , y − n, z − n )) and (( x , y , z − n ) , ( x , y , z + 1 − n )) . Thus they are at Euclidean distance ≥ n , so in particular (cid:96) ( v ( N − i ) − v (cid:48) ( N (cid:48) − i )) ≥ n . This shows that part (ii) of thedefinition of biautomaticity does not hold for L . (cid:3) References [Bah06] Patrick Bahls,
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Department of Mathematics and Statistics, McGill University, Burnside Hall, 805Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
E-mail address : [email protected] Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50–384 Wroc-ław, PolandInstitute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 War-szawa, Poland
E-mail address : [email protected] Department of Mathematics and Statistics, McGill University, Burnside Hall, 805Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
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