ORDER AUTOMATIC MAPPING CLASS GROUPS. Colin Rourke and Bert Wiest

Transcription

1 PACIFIC JOURNAL OF MATHEMATICS ol. 94, No., 000 ORDER AUTOMATIC MAPPING CLASS GROUPS Colin Rourke nd Bert Wiest We prove tht the mpping clss group of compct surfce with finite numer of punctures nd non-empty oundry is order utomtic. More precisely, the group is right-orderle, hs n utomtic structure s descried y Mosher, nd there exists finite stte utomton tht decides, given the Mosher norml forms of two elements of the group, which of them represents the lrger element of the group. Moreover, the decision tkes liner time in the length of the norml forms. 0. Introduction. This pper is sequel to [5] nd contins the proof of the liner time lgorithm which ws nnounced in [5, Remrk 5.]. In ddition to giving the full proofs, this pper extends the results of [5] to considerly lrger clss of mpping clss groups. Let S e compct surfce, not necessrily orientle, with oundry S nd finite numer of distinguished points in its interior, clled the punctures. The mpping clss group MCG(S) is defined to e the group of homeomorphisms of S mpping S identiclly nd permuting the punctures, up to isotopies fixing S nd the punctures. The group multipliction is given y composition nd written lgericlly, i.e., given Φ, Ψ MCG(S) define ΦΨ := Ψ Φ: S S. For instnce, if S is disk with n punctures then MCG(S) is isomorphic to B n, the rid group on n strings []. In [, 3] P. Dehornoy defined totl ordering on the rid group B n which is right invrint, i.e., for β, γ, δ B n, β>γimplies βδ > γδ. This ordering ws reinterpreted in [5] in more geometricl terms. Right invrint orderings re importnt ecuse of long-stnding conjecture in group theory: Tht the group ring of torsion-free group hs no zero divisors. The conjecture is true for the smller clss of right-orderle groups (groups dmitting right invrint order). For more detil see [8, Chpter ]. 09

2 0 COLIN ROURKE AND BERT WIEST The mpping clss group of surfce S is torsion-free if nd only if S is non-empty. In Section we generlise the construction in [5] to prove tht for ny surfce S with non-empty oundry, MCG(S) is right-orderle. In [6, 7] L. Mosher constructed n utomtic structure (in the sense of [4]) for mpping clss groups of surfces. More precisely, suppose tht surfce S with oundry nd possily some punctures is equipped with tringultion whose vertex set consists of the set of punctures nd some points in the oundry, nd whose edges re ordered nd oriented. Then Mosher ssocites with every element Φ of MCG(S) unique finite sequence of pirs of tringultions of S, which we cll the utomtic norml form of Φ. In Section nd 3 we construct finite stte utomton tht detects the order from the utomtic norml forms, i.e., given the utomtic norml forms of two elements, the utomton decides which element is lrger. The resulting lgorithm tkes liner time in the length of the norml forms; indeed the only unounded prt of the lgorithm is reding the sequences to detect the first discrepncy. If the two sequences of tringultions first differ in the ith term, then the order cn e red from the ith to i + 3rd elements of the sequences. Section contins the proof of weker result tht the lgorithm tkes qudrtic time in the length of the norml forms nd Section 3 contins the technicl detils which shrpen this to liner time. It is worth noting tht our results extend to sugroups of mpping clss groups defined y restricting the llowed permuttion of the punctures to lie in some sugroup of the permuttion group. It is lso worth oserving tht there is no greter generlity in considering homeomorphisms which re the identity on just some of the oundry components. This is ecuse oundry component which is not fixed is essentilly the sme s nother puncture.. A right invrint order on MCG(S). In this section we construct right invrint order on MCG(S). The construction involves lot of choices, nd there re mny other such orderings This result ppers to hve the sttus of folklore theorem. Although clerly known for long time nd implicit in erly work on mpping clss groups, we hve filed to locte n explicit reference. There re two well-estlished conventions for products in mpping clss groups, the lgeric convention used here nd the functionl convention ΦΨ := Φ Ψ. Using the functionl convention, the order constructed in this pper is left invrint. However there is no rel distinction etween left nd right invrint orders ecuse ny group dmitting left invrint order lso dmits right invrint order y g<h h g nd vice-vers.

3 ORDER AUTOMATIC MAPPING CLASS GROUPS prt from the ones we exhiit. The mteril in this section is strightforwrd generlistion of Sections -3 of [5]. We shll leve out some detils, nd refer the reder to [5] insted. Idel rc systems nd their reductions. Suppose S is compct surfce with non-empty oundry nd finite numer of punctures (possily none). The set of punctures is denoted p. Following [6] we define n idel rc to e the imge h of mp (I, I,int I) (S, S p, S ( S p)) which is injective in int I. Two idel rcs re isotopic if there exists n isotopy of S fixed on S p deforming one into the other. An idel rc system Γ is collection of non-isotopic idel rcs with disjoint interiors, such tht ech component of S Γ is disk. Two idel rc systems cn e reduced with respect to ech other or pulled tight. We shll sketch this, detils cn e found in [7] or[5]. Suppose Γ nd re two idel rc systems. We define Γ nd to e trnsverse if every rc of Γ either coincides precisely with some rc of, or intersects the rcs of trnsversely. Now suppose Γ nd re trnsverse. A D disk etween Γ nd is suset of S homeomorphic to closed disk which contins no punctures in its interior, nd which is ounded y one segment of n idel rc of Γ nd one of ; the two intersection points of the two idel rcs my e in the interior or in the oundry of the rcs. We sy two idel rc systems Γ 0,Γ re equivlent with respect to if there is n isotopy of S tht fixes S p, leves the suset S invrint, nd crries Γ 0 to Γ. For proofs of the following two fundmentl results see [7] or[5]: Proposition.. Any two idel rc systems Γ, cn e reduced with respect to ech other, i.e., fter n pproprite isotopy of Γ there re no D-disks etween Γ nd. Moreover, this reduction process gives unique result: If two curve digrms Γ 0, Γ re oth trnsverse to nd reduced with respect to, then they re equivlent with respect to. Proposition. (Triple reduction lemm). Suppose Γ, nd Σ re three idel rc systems such tht Γ nd re oth reduced with respect to Σ. Then there exists n isotopy etween Γ nd n idel rc system Γ, which is n equivlence with respect to Σ, such tht Γ, nd Σ re pirwise reduced. Curve digrms. Next we construct specil exmples of idel rc systems, with some dditionl lelling. We shll cll ny digrm on S which cn e otined in this wy curve digrm (s in [5]).

4 COLIN ROURKE AND BERT WIEST First we construct n idel rc system Γ of S with the following properties: All idel rcs re emedded nd disjoint (even in their endpoints), ll endpoints of idel rcs lie in S, nd S Γ consists of precisely one disk, possily punctured. If Γ hs sy k idel rcs, then we lel these rcs Γ,...,Γ k, in ny order. Moreover, we equip ech of them with n orienttion. Next, if S hs n punctures, then we lel these punctures,...,n,in ny order, nd dd to the digrm Γ n dditionl n emedded idel rcs with disjoint interiors s follows. We hve one oriented idel rc from S to the first puncture, lelled Γ k+, disjoint from ll the previous rcs; then one oriented idel rc from the first to the second puncture, nd so on up to the rc Γ k+n, from the n st to the nth puncture. This finishes the construction of generl curve digrm. For n exmple see Figure. Here S is torus with one disk removed nd three punctures (so k =,n= 3). Γ Γ Γ 3 Γ 4 Γ 5 Figure. A curve digrm on torus with one oundry component nd 3 punctures. We now fix, once nd for ll, curve digrm E on S, which we cll the sic curve digrm. The choice of E is ritrry, ut it will serve s sepoint for the constructions tht follow. We lso choose, once nd for ll, n orienttion for S. (Recll tht we re not ssuming tht S is itself orientle. Even if S is orientle, nothing tht follows depends on choice of orienttion for S.) The right invrint ordering of MCG(S). Curve digrms re the min uilding lock in the construction of right invrint ordering of MCG(S). If Φ is n element of MCG(S), then pplying Φ to E yields nother curve digrm Φ(E). Note tht Φ(E) is only determined up to isotopy. Given Φ, Ψ MCG(S), we cn ssume tht the curve digrms Φ(E) nd Ψ(E) re reduced with respect to ech other (recll tht Φ(E) nd Ψ(E)

5 ORDER AUTOMATIC MAPPING CLASS GROUPS 3 hve unique reduced form with respect to ech other, y Proposition.). Suppose tht the idel rcs Φ(E l ) nd Ψ(E l ) coincide for l =,...,i, where i k + n, nd tht Φ(E i ) nd Ψ(E i ) re the first non-coincident pir. We note tht Φ(E i ) nd Ψ(E i ) strt t the sme point of S or puncture, re oriented, nd tht oth Φ(E) nd Ψ(E) re loclly seprting. So it mkes sense to define Φ > ΨifΦ(E i ) rnches off Ψ(E i ) to the left, i.e., if n initil segment of Φ(E i ) lies to the left of Ψ(E i ). Here we re using the chosen orienttion of S to define left nd right. Left mens in the direction of the orienttion of S. Figure illustrtes this. S S Φ(E i ) S S Φ(E i ) Φ(E i ) Ψ(E i ) Ψ(E i ) Ψ(E i ) Figure. i {,...,k+} i=k+ The trnsitivity of this reltion follows from the triple reduction lemm: Suppose tht Φ, Ψ, Y MCG(S), then y the triple reduction lemm. we cn ssume tht the three curve digrms Φ(E), Ψ(E), Y(E) re pirwise reduced. Then if Y(E) rnches off Φ(E) to the left, nd Φ(E) rnches off Ψ(E) to the left, then Y(E) rnches off Ψ(E) to the left. So < is indeed n order. The fct tht this order is right invrint is lso immedite from the definitions: If Φ, Ψ, Y MCG(S), then in order to compre ΦY with ΨY, we hve to pply homeomorphism of S representing Y to the two digrms Φ(E) nd Ψ(E). This leves the digrms reduced with respect to ech other, nd therefore preserves the order.. Automtic ordering. In this section nd the next we prove tht the mpping clss group MCG(S) is order utomtic. More precisely, we construct n lgorithm tht is executle y finite stte utomton, hs s its input the utomtic norml forms (in the sense of Mosher [6, 7]) of two elements of MCG(S), nd s its output the decision which of the two elements is lrger. Furthermore, we prove tht the lgorithm tkes liner time in the length of the utomtic norml forms. A wek lgorithm (tking qudrtic time) is constructed in

6 4 COLIN ROURKE AND BERT WIEST this section. The next section contins the necessry technicl detil to shrpen this to liner time. Review of Mosher s norml form for elements of MCG(S). An idel tringultion of S is tringultion whose edges re idel rcs nd segments of S, nd whose vertices re punctures nd endpoints of idel rcs in S. Moreover, the non-oundry edges re oriented, nd numered,...,e, where e is the numer of non-oundry edges. We consider two idel tringultions to e equivlent if they differ y oundry- nd puncture-fixing isotopy. We sy two idel tringultions re in the sme tringultion clss if they re comintorilly equivlent reltive to the oundry, or equivlently, if they differ y the ction of n element MCG(S). Note tht the edges nd the tringles of the tringultions eing considered re llowedtoeent, i.e., edges my hve oth ends coincident nd tringles my hve one or more edges or vertices coincident. Suppose now tht S is equipped with n idel tringultion B, which we cll the se tringultion. We consider only tringultions which gree with B on S. We now consider the groupoid G which hs for ojects the set of tringultion clsses nd for morphisms the set of ordered pirs (T,T ) of idel tringultions, where (T,T ) is identified with (h(t ),h(t )) if h MCG(S). The morphism goes from the clss of T to the clss of T. If T nd T re in the sme clss, then there is unique oundry fixing isomorphism from T to T up to isotopy, i.e., n element of MCG(S). This determines n isomorphism etween the vertex group of G nd the mpping clss group MCG(S) with multipliction now written functionlly i.e., ΦΨ := Φ Ψ. We consider prticulr type of morphism in G. Definition (Flipping n edge). Let T e n idel tringultion nd n edge of T djcent to two tringles δ nd δ. The tringultion T is otined y removing, so tht δ nd δ comine to form qudrilterl of which is digonl, nd then cutting the qudrilterl ck into two tringles y inserting the opposite digonl. We cll the morphism (T,T ) flipping nd denote it f, see Figure 3. f Figure 3. Flipping n edge. Every morphism (B,T) ingfrom the se vertex to nother vertex is product of cnonicl sequence of flips. To see this, picture (B,T) s given

7 ORDER AUTOMATIC MAPPING CLASS GROUPS 5 y superimposing B nd T, nd com T long B. To e precise, first reduce T with respect to B nd then consider edge of B. Suppose tht the first edge of T tht edge crosses is. Flip s in Figure 3, where edge is drwn dshed, nd reduce. Repet this process until there re no more crossings of edge with T. The fct tht this process is finite follows y counting the numer of interections of edge with T (excluding the nextto-e-flipped edge). For more detil here see [7, pges 3 3]. Now do the sme for edge nd continue in this wy, using the chosen ordering nd orienttion of edges of B, until T hs een converted into copy of B. The Mosher norml form of (B,T) is the inverse of the sequence of flips descried ove. 3 Notice tht unlike the generl cse descried in [7] there is no relelling morphism required here. (This is ecuse S is fixed throughout.) Mosher proves tht this norml form defines n utomtic structure on G nd hence, using results from [4], on the vertex group MCG(S). Now consider n element (B,T) of the vertex group t the clss of B. As remrked erlier (B,T) determines n element Φ MCG(S) unique up to isotopy such tht Φ(T ) is equivlent to B. (Conversely given Φ MCG(S) the corresponding tringultion pir is (B,Φ B).) We oserve tht if we com B long Φ(B) this is comintorilly identicl to coming Φ (B) long B nd we cll the sequence of flips defined y this coming the coming sequence of Φ. (The reverse of the coming sequence is the Mosher norml form of Φ.) The se tringultion. For our purposes we need se tringultion B with certin specil properties. We shll construct B from the sic curve digrm E in two steps. In the first step we construct n idel rc system which hs three edges B 3i,B 3i,B 3i for every rc E i of E. This step needs cre; y contrst, the second step is quite unsutle: We simply dd some more edges, to turn the rc system into n idel tringultion. For exmple, Figure 4 shows se tringultion otined from the sic curve digrm in Figure. Here the edges constructed in Step re drwn with solid lines nd the edges constructed in Step with dshed lines. Step. We recll tht E hs k idel rcs E,...,E k with endpoints in S, nd n rcs E k+,...,e k+n with puncture t t lest one of their endpoints. We define B 3i = E i for i =,...,k + n. Here B 3i lso 3 Strictly speking the Mosher norml form is not this flip sequence, which only defines n synchronous utomtic structure, ut is derived from it y clumping flips together into locks clled Dehn twists, prtil Dehn twists nd ded ends (see [7] pges 34 et seq.). This techniclity does not ffect ny of the results proved here. We prove tht order cn e detected in liner time from the flip sequence. Since the clumped flip sequence cn e unclumped in liner time, this implies tht order cn e detected in liner time from the strict Mosher norml form.

8 6 COLIN ROURKE AND BERT WIEST crries the sme orienttion s E i nd we lel the strting point of E i s i for future reference. For i =,...,k+ we choose two points of S on either side of i nd closer to i thn ny other oundry vetex. We define the edge B 3i to strt t the finishing point of E i, run on the left of nd close to E i, ut with opposite orienttion, nd end on S, ner the strting point of E i,t one of the points just chosen. The edge B 3i is defined to sit on the other side of E i = B 3i, in similr fshion. So B 3i,B 3i,B 3i nd segment of S together ound two simplices, with common edge B 3i. We cll this ensemle Q of two simplices with edges B 3i,...,B 3i the ith ek of B. The edge B 3i for i = k +,...,k+n is defined to strt t the i kth puncture, run close to B 3i nd B 3i 4, nd to end t the sme point of S s B 3k+. So the edge B 3i sits to the left of B 3i. The edge B 3i for i = k +,...,k+n re defined similrly, ut on the right of B 3i. So the edges B 3i 4,...,B 3i together ound two simplices, nd gin we cll this ensemle the ith ek. Let Q i e the ith ek, i {,...,k+n}. We cll the edge B 3i the leding edge of Q i, nd B 3i nd B 3i the left nd right outlying edges of Q i. We lso cll the strting point i of the leding edge the principl vertex of Q i. For exmple, in Figure 4, the 5th ek Q 5 is shded nd hs principl vertex 5, leding edge 3 nd outlying edges 4 nd 5. S 3 S 4 5 Q S S Figure 4. A se tringultion otined from the sic curve digrm in Figure.

9 ORDER AUTOMATIC MAPPING CLASS GROUPS 7 Step. In generl, the idel rc system defined in Step is not n idel tringultion of S, ecuse some of the cells re not tringles. We now dd finite numer of idel rcs so s to crete n idel tringultion. Orienttion or lelling of these extr edges will not ply ny role in the proof, nd we shll think of them s unoriented nd unlelled. Detecting order from the Mosher norml form. We re now redy to stte the min result of this section. Theorem. (Min Theorem). There exists n lgorithm tht hs s its input the coming sequences of two elements Φ,Ψ of MCG(S), nd s its output the decision which of the two elements is lrger. Furthermore, if the two coming sequences of Φ,Ψ first differ in the ith term, then the order cn e red y inspecting the ith to i + 3rd terms of the sequences nd consulting finite ctlogue of possiilities. Notice tht the theorem implies tht decision tkes liner time in the length of the shortest coming sequence, indeed the only unounded prt of the process is the inspection of the two sequences to locte the first discrepncy. Furthermore the decision cn clerly e crried out y finite-stte utomton which hs the necessry ctlogue uilt in. Thus we hve: Corollry.. The mpping clss group MCG(S) is order utomtic. Before strting on the detiled proof of the theorem it is worth explining the strtegy. The reltive order of two elements Φ,Ψ of MCG(S) is determined y the imges of the curve digrms, which re prt of the se tringultion. The coming sequences re otined y coming long the imges of the se tringultion so it is plusile tht the position of the curve digrm up to certin point cn e reconstructed from the coming sequence up to the corresponding point. However there re prolems. Coming long given edge of the curve digrm my not cuse ny flips in the coming sequence for the simple reson tht the edge is lredy prt of the tringultion t tht point. But notice tht we hve chosen the tringultion so tht ech edge in the curve digrm is the leding edge of ek nd the corresponding outlying edges cn e used to determine the position of the leding edge in this cse. There re further complictions due to the fct tht, in the middle of the sequence, the tringultion which is eing comed my well hve ent edges or tringles. If ent edge is flipped it is often miguous where the edge which cused the flip is locted. The whole process cn e likened to oserving ule chmer (the ules re the flips) nd trying to work out the position of the prticles (edges which we re coming long) which cused the ules. Remrk.3. There is very short proof of somewht weker result, long the lines of the proof given in [5]. By inspecting the coming sequence

10 8 COLIN ROURKE AND BERT WIEST we cn immeditely red whether n element of MCG(S) is positive (i.e., greter thn the identity element) or negtive. More precisely, define Φ MCG(S) to e i-positive (respectively i-negtive) if the first i curves of Φ(E) coincide (fter reduction) with the first i ofend tht Φ(E i ) rnches off E i to the left (respectively right). Now the curve digrm Φ(E) of Φ is prt of the tringultion Φ(B) nmely the edges numered, 4, 7,... Suppose tht Φ is i positive, then it cn e ssumed to fix the first i of these edges (i.e.,, 4,...,3i 5) nd, fter reduction, the corresponding outlying edges (i.e.,, 5,...,3i 4 nd 3, 6,...,3i 3). But edge 3i is crried to the left nd must meet edge 3i ofb. Thus the first flip in the coming sequence of Φ is f 3i, i.e., flip of the edge numered 3i. Similrly if Φ is i negtive then the first flip in the coming sequence is f 3i. We hve proved the following: Algorithm.4. (To decide from the Mosher norml form whether n element of MCG(S) is i-positive or negtive nd provide the correct vlue of i.) Inspect the coming sequence (the reverse of the Mosher norml form). The first flip is either f 3i or f 3i for some i. In the first cse the element is i- positive nd in the second it is i-negtive. From generl properties of utomtic structures (see [4, Theorem.3.0]), the coming sequence of cn e computed in qudrtic time from the coming sequences of nd, hence Algorithm.4 gives qudrtic time lgorithm to determine order nd implies Corollry.. The finl section of the pper comprises the technicl detil necessry to shrpen this into liner time lgorithm. 3. Proof of the min theorem. We strt y oserving tht in order to detect order we only need to exmine the coming of the ek prt (i.e., the first 3(k +n) rcs of the tringultion creted in Step ove). If coming long the ek prt of Φ(B) nd Ψ(B) gives the sme result, then the curve digrms Φ(E) nd Ψ(E) gree, so tht Φ = Ψ MCG(S), nd the complete coming sequences of Φ nd Ψ coincide. So in order to compre Φ nd Ψ, we need not com long the edges creted in step ove (nd this is the reson why their lelling nd orienttion is irrelevnt). Now consider definite coming sequence f,f,...f i,f i+,f i+,... nd consider the tringultion T t the strt of f i. Assume tht f i comes from coming long prticulr ek Q. The principl vertex, leding edge etc of Q will e clled the principl vertex etc t tht stge.

11 ORDER AUTOMATIC MAPPING CLASS GROUPS 9 As the coming process proceeds we construct sequence of eks which coincide with eks of Φ(B). These eks re never touched gin throughout the coming process nd we cll the union of the eks constructed thus fr the stle set, denoted U. Oservtion 3.. At ny stge in the coming process we cn identify the stle set nd hence the principl vertex. Indeed the next edge to e flipped is lwys in the str of (i.e., the collection of tringles of which is vertex, together with their fces), where is the principl vertex. So we look for the mximl set of eks of the tringultion T t tht stge which contin vertices,,..., j such tht the next flip is not in the str of i for i<j. This is the stle set nd = j. The next flip is cused y coming long some ek Q. The leding edge q of Q is either () lredy n edge of T (in this cse we re coming long one of the outlying edges of Φ(B)) or () crosses tringle A of T from the leding vertex to the opposite edge (which is the next edge flipped). If we cn determine q in Cse () or A in Cse (), then we sy tht we hve locted the leding edge q. Lemm 3. (Min technicl lemm). We cn locte q y inspecting T nd the four flips f i,f i+,f i+,f i+3 nd consulting finite ctlogue of possiilities. The min theorem follows from the lemm s follows. If the coming sequences of Φ nd Ψ first differ in the ith term then the leding edges t stge i must differ. This is ecuse the outlying edges re determined y the leding edge (they follow the leding edge ck nd out s indicted in Figure 5). Thus if the leding edges of the two eks coincide then so do the entire eks nd coming long the current eks would not cuse discrepncy. S or U Figure 5. The outlying edges re close to the leding edge. Now suppose tht the two principl vertices differ. For definiteness suppose tht Φ nd Ψ hve principl vertices k nd l respectively, with k<l

12 0 COLIN ROURKE AND BERT WIEST (note tht k nd l cn e determined y Oservtion 3.). Then T contins ek with principl vertex k nd leding edge q sy which is lredy prt of Ψ(B) ut not of Φ(B). Thus there is only one edge of T not in the stle set of Φ emnting from k, nd this must e q. Further the leding edge for Φ must e different. Thus y locting the ltter we cn red off the order. Now suppose tht the two principl vertices re the sme. Then oserve tht the two leding edges cnnot oth cross the sme tringle (or else the next flip f i would e the sme in oth sequences). Thus y locting the two leding edges we cn red off the reltive order nd the theorem is proved. Proof of the min technicl lemm. We shll show how to locte the leding edge from T nd the four flips. It will e cler tht the decision is equivlent to consulting finite ctlogue. There re two possiilities. We cn e coming long the leding edge or long one of the outlying edges. We suppose first tht we know which of these possiilities holds nd we del with them in cses A nd B. In cse C we del with the generl cse, which includes the possiility tht we my not immeditely know whether we re in cses A or B. Cse A: Coming long the leding edge. By Oservtion 3. we cn red off the principl vertex, sy. Since we re coming long the leding edge, the first flip f i is the flip of n edge such tht there is tringle A with vertex nd opposite edge. If there is only one such tringle, then the leding edge must cross this tringle nd we hve locted it. However there re numer of situtions where this tringle is not unique, the simplest of which is illustrted in Figure 6. (In this nd lter digrms the letter indictes tht the tringultion is continued in some wy in the mrked region.) A 3 4 Figure 6. Cse A: Stndrd miguity. There re two possile positions for the leding edge, lelled nd in the figure. But if the leding edge is in position then the second flip f i+

13 ORDER AUTOMATIC MAPPING CLASS GROUPS must e edge 3 or 4 nd in position it must e or. Thus we cn resolve the miguity nd locte the leding edge y inspecting the second flip. There is degenerte version of Cse A, when edges nd 3 coincide, Cse A in Figure 7. In this cse edge cnnot e the second flip for either A =3 4 Figure 7. Cse A: Degenerte version of Cse A. possile position of the leding edge, ecuse otherwise the leding edge would self-intersect; thus if the leding edge is in position then the second flip must e edge 4, nd in position it must e. Next come three ent versions, Cses A3, A4 nd A5 (Figure 8). In Cse A3 the inner end of from Cse A coincides with nd in Cse A4 the outer end. In Cse A5 the outer end of from Cse A coincides with. There is no cse where the inner end from Cse A coincides with since this would produce -gon. A3 A4 A =3 4 4 Figure 8. Cses A3, A4 nd A5: Bent version of A nd A. In Cses A3 nd A4 the miguity is resolved exctly s in Cse A, nd in Cse A5 the miguity is resolved exctly s in Cse A. In Cse A6 (Figure 9) oth ends of from Cse A hve een deformed to (nd the miguity is gin resolved s in Cse A) nd finlly to complete the ctlogue of cses where two different tringles hve the sme vertex nd se we hve included Cse A7 (Figure 9). There is no position for

14 COLIN ROURKE AND BERT WIEST the leding edge in Cse A7 so this cse cnnot occur; (the two different tringles re in fct the sme tringle in two different guises). A6 A7 3 4? Figure 9. Cses A6 nd A7: ery ent versions. Cse B: Coming long n outlying edge. We use the sme nottion s in Cse A: The principl vertex is nd the first flip f i is the flip of n edge. This time however must hve one endpoint t (cf. Figure 5). First ssume tht is stright (i.e., not ent). Cse B (Figure 0) is the unmiguous cse. This is the cse where there re either or no edges with vertex trpped to the left of etween nd the stle set. The leding edge must e the next edge to the right of. Cse B (Figure 0) is the miguous cse. This is the cse where there is precisely one edge (lelled ) with vertex trpped to the left of nd the leding edge could e this edge ( is flipped y the right outlying edge) or the edge lelled to the right of, if such n edge exists (if not this cse is lso unmiguous: The leding edge is ). However if is the leding edge then the second flip is edge (nd vice-vers); thus the second flip resolves the miguity. B or no edges B leding edge Figure 0. Cses B nd B: stright. Next ssume tht is ent. In this cse (Figure ) there re three possile positions for the leding edge, lelled, nd c in the figure

15 ORDER AUTOMATIC MAPPING CLASS GROUPS 3 (Cse B3) note tht is possile only if there is precisely one edge to the left of, nd tht nd c re the next edges to the right of the two rnches of, ssuming tht c exists. Assume first tht there is t lest one edge to the left of. If the leding edge is in positions or c, then the second flip is of n edge inside, whilst in position the second flip is outside. Thus the second flip distinguishes position. Finlly positions nd c re distinguished y the second flip (inside the flip is next to the left side for nd next to the right side for c) except in the specil cse B4. Here there is just one edge inside, which is the second flip for oth positions nd c; however in this cse the third flip f i+ distingushes the cses (it is edge for leding edge in position c nd vice-vers). B3 B4 c c Figure. Cses B3 nd B4: ent, t lest one edge to the left of. Now ssume tht there re no edges to the left of (Cses B5, B6 nd B7 in Figure ). The possile positions for the leding edge re nd c (ssuming c exists). Cse B5 is the cse when there re t lest two edges inside to the right of. In this cse the second flip distingushes nd c; it is the edge closest to (edge ) if the leding edge is in position nd the edge closest to (edge ) for c. Cses B6 nd B7 show the specil cses when there re one or no edges (respectively) inside to the right of. In Cse B6 the third flip distingushes (it is for position c nd vice-vers) nd in Cse B7 the second flip distingushes (it is gin for position c nd vice-vers). Cse C: The generl cse. Finlly we turn to the generl cse. If the first flip f i is cused y coming long the leding edge, then the edge which is flipped is n edge of tringle A with opposite vertex the principl vertex. If f i is cused y coming long one of the outlying edges then hs one endpoint t.thus unless stisfies oth these conditions we cn immeditely decide whether we re in Cse A or B. However, if the tringle A is ppropritely ent, it is possile for to stisfy oth conditions. The three possile shpes re illustted in Figure 3.

16 4 COLIN ROURKE AND BERT WIEST B5 B6 B7 c c c Figure. Cses B5, B6 nd B7: ent, no edges to the left of. A A A Figure 3. Cse C: Three possile shpes for A. One of the edges of A other thn must e ent. The possiilities re tht is outside this ent edge (left in Figure 3), inside (middle) or lso ent (right in this cse cn e ny of the three edges of A). Before discussing ech of these cses in detil it is worth remrking tht the miguous cses for coming the leding edge (Cses A to A7) cnnot occur here ecuse in these cses it cn e checked tht there is no position for the leding edge so tht n outlying edge will flip. Thus we never meet the cse A miguity together with miguity etween Cses A nd B. We now discuss ech of the three shpes in Figure 3 in turn. Figure 4 shows the two possile orienttions for the cse when is outside the ent edge of A. In Cse C the possile positions for the leding edge re nd nd the second flip distinguishes them (it is edge for position ut not for ). In Cse C the two possile positions for the leding edge re gin nd nd the second flip for is edge. This my lso e the second flip for position in the cse tht does not cross ny more edges of T, ut in this cse the third flip distinguishes the two posiilities. It is the leftmost edge inside (lelled 3) for nd the rightmost (lelled 4) for. There is one specil cse (like Cse B4 ove) when 3 nd 4 coincide nd this is Cse C3 shown in Figure 5.

17 ORDER AUTOMATIC MAPPING CLASS GROUPS 5 C C 3 4 Figure 4. Cses C nd C: outside. C3 3 Figure 5. Cse C3: The worst cse. In Cse C3 the leding edge in position does not cross ny more edges of T nd the first three flips for position nd coincide. However the fourth flip distinguishes the cses it is for position nd for position. This cse (nd the ent version, Cse C9 elow) is the only cse when it is necessry to inspect the fourth flip f i+3. Figure 6 shows the only possile orienttion for the cse when is inside the ent edge of A (the middle cse in Figure 3). Cse C4 shows the generl cse, nd Cse C5 the specil cse when the leding edge in position coincides with edge. In oth cses the second flip distinguishes it is for leding edge in position, ut not in position. Figure 7 shows the two possiilities for the totlly ent cse (the right hnd cse of Figure 3) when is inside. In oth cses the second flip distinguishes the possile positions for the leding edge in Cse C6 it is for position ut not for, nd in Cse C7 it is for position ut not for. Finlly Cse C8, Figure 8, shows the lst possiility for the totlly ent cse when is outside. In this cse there re three possile positions for the leding edge (lelled, nd c). The second flip detects c (the flip is for c ut not for nd ) nd finlly distinguishing etween nd is

18 6 COLIN ROURKE AND BERT WIEST C4 C5 = Figure 6. Cses C4 nd C5: inside. C6 C7 Figure 7. Cses C6 nd C7: ent nd inside. exctly the sme s in Cse C ove (indeed the figure is otined from C y deforming the top end of round to the right to coincide with ). In prticulr there is deformed copy of the worst Cse C3 (drwn s Cse C9) with identicl nlysis. C8 C9 c 3 Figure 8. Cses C8 nd C9: ent nd outside. Acknowledgement. Bert Wiest is supported y TMR (Mrie Curie) reserch trining grnt.

19 ORDER AUTOMATIC MAPPING CLASS GROUPS 7 References [] J. Birmn, Brids, links, nd mpping clss groups, Annls of Mth. Studies, 8, Princeton University Press, Princeton, 975. [] P. Dehornoy, Brid groups nd left distriutive opertions, Trns. AMS, 343 (994), [3], A fst method of compring rids, Adv. in Mth., 5 (997), [4] D.B.A. Epstein et l, Word processing in groups, Jones & Brtlett, 99. [5] R. Fenn, M.T. Greene, D. Rolfsen, C. Rourke nd B Wiest, Ordering the rid groups, Pcific J. Mth., 9() (999), [6] L. Mosher, Mpping clss groups re utomtic, Mth. Reserch Letters, (994), [7], Mpping clss groups re utomtic, Annls of Mth., 4 (995), [8] D.S. Pssmn, Algeric structure of group rings, Wiley, 977. Received Mrch 9, 998. University of Wrwick Coventry, C4 7AL United Kingdom E-mil ddress: cpr@mths.wrwick.c.uk University of British Columi ncouver BC, 6T Z Cnd E-mil ddress: ertw@pims.uc.c This pper is ville vi