A finite presentation for the twist subgroup of the mapping class group of a nonorientable surface
AA FINITE PRESENTATION FOR THE TWISTSUBGROUP OF THE MAPPING CLASS GROUP OF ANONORIENTABLE SURFACE
MICHA(cid:32)L STUKOW
Abstract.
Let N g,s denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski[12] obtained an explicit finite presentation for the mapping classgroup M ( N g,s ) of the surface N g,s , where s ∈ { , } and g + s >
3. Following this work, we obtain a finite presentation for thesubgroup T ( N g,s ) of M ( N g,s ) generated by Dehn twists. Introduction
Let N g,s be a smooth, nonorientable, compact surface of genus g with s boundary components. If s is zero, then we omit it from the notation.If we do not want to emphasise the numbers g, s , we simply write N for a surface N g,s . Recall that N g is a connected sum of g projectiveplanes and N g,s is obtained from N g by removing s open disks.Let Diff( N ) be the group of all diffeomorphisms h : N → N suchthat h is the identity on each boundary component. By M ( N ) wedenote the quotient group of Diff( N ) by the subgroup consisting ofmaps isotopic to the identity, where we assume that isotopies are theidentity on each boundary component. M ( N ) is called the mappingclass group of N .The mapping class group M ( S g,s ) of an orientable surface is definedanalogously, but we consider only orientation preserving maps.1.1. Background.
One of the most important elements in mappingclass groups of surfaces are Dehn twists. They were discovered by MaxDehn, who first observed that they generate the mapping class group M ( S g ) of a closed oriented surface S g . Twists were rediscovered byLickorish [8, 10], who also proved that M ( S g ) is generated by 3 g − Mathematics Subject Classification.
Primary 57N05; Secondary 20F38,57M99.
Key words and phrases.
Mapping class group, Nonorientable surface, Twist sub-group, Presentation.Supported by NCN grant 2012/05/B/ST1/02171. a r X i v : . [ m a t h . G T ] A ug MICHA(cid:32)L STUKOW
Dehn twists about nonseparating circles. Later Humphries reducedthis generating set to 2 g + 1 twists [4].Since Dehn twists generate the mapping class group M ( S g ), it isnatural to ask about possible relations between them. Let us men-tion some results in this direction. Birman [1] observed that there isa close relation between mapping class group M ( S g ) and the mappingclass group of a punctured sphere, which in fact is a quotient of thebraid group B g +2 . This correspondence leads to a number of inter-esting relations, for example: braid and chain relations , relations withhyperelliptic involution, relations with elements of finite order. LaterJohnson [5] discovered the so-called lantern relation , which apparentlyhas been used by Dehn in 1920’s. It turned out that this set of re-lations was enough to give a full presentation of M ( S g ), which wasobtained by Wajnryb [16]. Later some other relations were discovered,for example star relations or relations between fundamental elementsin Artin groups embedded in M ( S g ). These relations led to some otherinteresting presentations of M ( S g ) – see [3, 11].In the nonorientable case, Lickorish [9] first observed that Dehntwists do not generate the mapping class group M ( N g ) for g ≥ twistsubgroup T ( N g ) which is of index 2 in M ( N g ). Later Chillingworth [2]found finite generating sets for T ( N g ) and M ( N g ). These generat-ing sets were extended to the case of a surface with punctures and/orboundary components in [6, 13, 14].As for relations, recently Paris and Szepietowski [12] obtained a finitepresentations for groups M ( N g,s ) where s ∈ { , } and g + s > Main results.
The main goal of this paper is to find a completeset of relations between Dehn twists on a nonorientable surface N .To be more precise, we obtain a presentation for the twist subgroup T ( N g,s ) of the mapping class group M ( N g,s ) of a nonorientable surface(Theorems 3.1 and 3.2), where s ∈ { , } and g + s >
3. The ob-tained presentations may seem to be complicated, but many relationsare needed only for small genera and stably the presentations are quitesimple.Our starting point is the presentation of M ( N g,s ) obtained by Parisand Szepietowski [12], however their presentation has g − T ( N g,s ), hence it leads to a very complicatedpresentation of the twist subgroup. Therefore, we use a recent sim-plification of their presentation [15], which has only one generator notbelonging to T ( N g,s ) (Theorems 2.1, 2.2 and 2.3). FINITE PRESENTATION FOR THE TWIST SUBGROUP. . . 3 Preliminaries
Notation.
Let us represent surfaces N g, and N g, as respectivelya sphere or a disc with g crosscaps and let α , . . . , α g − , β be two-sided circles indicated in Figure 1. Small arrows in this figure indicate Figure 1.
Surface N as a sphere/disc with crosscaps.directions of Dehn twists a , . . . , a g − , b associated with these circles.Observe that β (hence also b ) is defined only if g ≥
4. From now onwhenever we use b , we silently assume that g ≥ µ and oriented two-sided circle α which intersects µ in one point (Figure 2), we define a crosscap slide (or Y-homeomorphism) Y µ,α , that is the effect of pushing µ along the curve α – for precise definition see Section 2.2 of [12]. In Figure 2.
Crosscap slide.particular, let y = Y µ ,α , where µ , α are curves indicated in Figure3. The following three theorems are the main results of [15] Theorem 2.1. If g ≥ is odd or g = 4 , then M ( N g, ) admits a pre-sentation with generators a , . . . , a g − , y and b for g ≥ . The definingrelations are (A1) a i a j = a j a i for g ≥ , | i − j | > , (A2) a i a i +1 a i = a i +1 a i a i +1 for i = 1 , . . . , g − , (A3) a i b = ba i for g ≥ , i (cid:54) = 4 , MICHA(cid:32)L STUKOW
Figure 3.
Circles µ i and α i .(A4) ba b = a ba for g ≥ , (A5) ( a a a b ) = ( a a a a b ) for g ≥ , (A6) ( a a a a a b ) = ( a a a a a a b ) for g ≥ , (B1) y ( a a a a ya − a − a − a − ) = ( a a a a ya − a − a − a − ) y for g ≥ , (B2) y ( a a y − a − ya a ) y = a ( a a y − a − ya a ) a , (B3) a i y = ya i for g ≥ , i = 3 , , . . . , g − , (B4) a ( ya y − ) = ( ya y − ) a , (B5) ya = a − y , (B6) byby − = [ a a a ( y − a y ) a − a − a − ][ a − a − ( ya y − ) a a ] for g ≥ , (B7) ( a a a a a a a a ya − a − a − a − a − a − a − a − ) b = b ( a a a a a a a a ya − a − a − a − a − a − a − a − ) for g ≥ , (B8) [( ya − a − a − a − ) b ( a a a a y − )][( a − a − a − a − ) b − ( a a a a )] =[( a − a − a − ) y ( a a a )][ a − a − y − a a ][ a − ya ] y − for g ≥ .If g ≥ is even, then M ( N g, ) admits a presentation with genera-tors a , . . . , a g − , y , b and additionally b , b , . . . , b g − . The definingrelations are relations (A1)–(A6), (B1)–(B8) above and additionally (A7) b = a , b = b , (A8) b i +1 = ( b i − a i a i +1 a i +2 a i +3 b i ) ( b i − a i a i +1 a i +2 a i +3 ) − for ≤ i ≤ g − , (A9a) b b = bb for g = 6 , (A9b) b g − a g − = a g − b g − for g ≥ . (cid:3) Theorem 2.2. If g ≥ , then the group M ( N g, ) is isomorphic to thequotient of the group M ( N g, ) with presentation given in Theorem 2.1obtained by adding a generator (cid:37) and relations (C1a) ( a a · · · a g − ) g = (cid:37) for g odd, (C1b) ( a a · · · a g − ) g = 1 for g even, (C2 ) (cid:37)a = a (cid:37) , FINITE PRESENTATION FOR THE TWIST SUBGROUP. . . 5 (C3) (cid:37) = 1 , (C4a) ( y − a a · · · a g − ya a · · · a g − ) g − = 1 for g odd, (C4b) ( y − a a · · · a g − ya a · · · a g − ) g − y − a a · · · a g − = (cid:37) for g even. (cid:3) Theorem 2.3.
Relations (C4a), (C4b) and (C2 ) in the presentationgiven by Theorem 2.2 may be replaced by (C2) (cid:37)a i = a i (cid:37) for i = 1 , . . . , g − , (C5) y(cid:37) = (cid:37)y − , (C4) ( y(cid:37)a a · · · a g − ) g − = 1 . (cid:3) Presentation for the twist subgroup
Recall that for s ≤ g ≥ T ( N g,s ) has index2 in M ( N g,s ) (for details see [9, 13]), hence we can obtain its presenta-tion using Reidemeister–Schreier rewriting process. To be more precise,we define a Schreier transversal U = { , y } for T ( N g,s ) in M ( N g,s ) andfor any h ∈ M ( N g,s ) we define h = (cid:40) h ∈ T ( N g,s ) y if h (cid:54)∈ T ( N g,s ).The Reidemeister–Schreier theorem states that T ( N g,s ) admits a pre-sentation with generators uxux − , where x is a generator of M ( N g,s ), u ∈ U and ux (cid:54)∈ U . The set of defining relations consists of relations ofthe form uru − , where u ∈ U and r is a defining relation for M ( N g,s ). Theorem 3.1. If g ≥ is odd or g = 4 , then T ( N g, ) admits a pre-sentation with generators a , . . . , a g − , e, f, y and b, c for g ≥ . Thedefining relations are (A1)–(A6) and (A1 ) ea j = a j e for g ≥ , j ≥ , (A1 ) f a j = a j f for g ≥ , j ≥ , (A2 ) a ea = ea e , (A2 ) a − ea − = ea − e for g ≥ , (A2 ) a f a = f a f , (A3 ) a c = ca for g = 4 , , (A3 ) ec = ce for g = 4 , , (A4) ca c = a ca for g = 5 , , (A5) ( e − a a c ) = ( a − e − a a c ) for g = 5 , , (A6) ( e − a a a a c ) = ( a − e − a a a a c ) for g = 7 , , (B1) ( a a a a ea a − e )( a a a a f a a − f ) = 1 for g ≥ , (B2 ) y = a a ea a a a a a f a a , (B2 ) ( a a ea a a a a a f a a )( a a f a a a a a a ea a ) = 1 , MICHA(cid:32)L STUKOW (B3) y a = a y for g ≥ , (B4 ) ea = a e , (B4 ) f a = a f , (B6 ) bc = [ a a a f − a − a − a − ][ a − a − e − a a ] for g ≥ , (B6 ) c ( y by − ) = [ a − e − a a a − ea ][ ea − ( y a y − ) a e − ] for g =4 , , (B7 ) ( a a a a a a a a ea a − ea − a − a − a − ) c = b ( a a a a a a a a ea a − ea − a − a − a − ) for g ≥ , (B7 ) ( a − a − a − a − a − a − a − a − ) b ( a a a a a a a a ) y = y ( a − a − a − a − a − a − a − a − ) b ( a a a a a a a a ) for g ≥ , (B8 ) (cid:2) ( a ea − a − ) c ( a a e − a − ) (cid:3) (cid:2) ( a − a − a − a − ) b − ( a a a a ) (cid:3) = a − (cid:2) ( a − a − e − a ) a ( a − ea a ) (cid:3) a − e − for g ≥ , (B8 ) (cid:2) ( a − a − a − a − ) b ( a a a a ) (cid:3) (cid:2) ( a f a − a − ) y − c − y ( a a f − a − ) (cid:3) = a − (cid:2) ( a − f a a ) a ( a − a − f − a ) (cid:3) f a for g = 5 , .If g ≥ is even, then T ( N g, ) admits a presentation with generators a , . . . , a g − , e, f, y , b, c and additionally b , b , . . . , b g − , b g − , b g − , b g − .The defining relations are relations (A1)–(A9), (A1 )–(A6), (B1)–(B8 ) and additionally (A7a) b = a − , b = c for g = 6 , (A7b) b = c for g = 8 , (A7c) b i = z g − b i z − g − where i = g − , g − , i ≥ and z g − = ( a g − a g a g − a g − · · · a a e − a a − e − )( a − a − · · · a − g − a − g − a − g a − g − ) , (A8a) b = ( b e − a a a b ) ( b e − a a a ) − for g = 6 , (A8b) b g − = ( b g − a g − a g − a g − a g − b g − ) ( b g − a g − a g − a g − a g − ) − for g ≥ , (A9a) b c = cb for g = 6 , (A9b) b g − a g − = a g − b g − for g ≥ .Proof. As noted before, we apply Reidemeister–Schreier theorem to thepresentation given by Theorem 2.1. Hence as generators of the twistsubgroup T ( N g, ) we obtain a , . . . , a g − , ya y − , . . . , ya g − y − , y and b, yby − for g ≥
4. Moreover, if g ≥ b , b , . . . , b g − , yb y − , yb y − , . . . , yb g − y − . Let us namesome of these generators: e = ya − y − , c = yby − , b i = yb i y − for i = 0 , . . . , g − . We also add one generator f = y − a − y with defining relation(D1) f = y − ey FINITE PRESENTATION FOR THE TWIST SUBGROUP. . . 7 (see Figure 4). (B3)
Observe first that relation (B3) rewrites as
Figure 4.
Twists e, f, y , c . ya i y − = a i for i = 3 , , . . . , g − . This means that we can remove generators ya y − , . . . , ya g − y − fromthe presentation, hence from now on we will silently identify ya i y − with a i for i = 3 , , . . . , g − (B5) Similarly, (B5) allows us to identify ya y − with a − .Observe also that conjugations of (B3) and (B5) by y give(B3) y a i = a i y for i = 1 , , , . . . , g − ) if i (cid:54) = 3. (A1)–(A9) Relations which do not contain y , that is (A1)–(A9) doesnot need rewriting, however we need to add their versions conjugatedby y . This gives relations (A1 ), (A2 ), (A2 ), (A3 ), (A4)–(A6) and(A3 ) a i c = ca i for g ≥ i (cid:54) = 2 , g ≥ b = a − , b = c ,(A8a) b = ( b e − a a a b ) ( b e − a a a ) − ,(A8b) b i +1 = ( b i − a i a i +1 a i +2 a i +3 b i ) ( b i − a i a i +1 a i +2 a i +3 ) − for 2 ≤ i ≤ g − ,(A9a) b c = cb for g = 6,(A9b) b g − a g − = a g − b g − for g ≥ (B4) Relation (B4) and its conjugation by y − rewrite respectively as(B4 ) and (B4 ). It is also useful to note that relations (D1), (B3),(A2 ) and (A2 ) imply that(A2 ) a f a = f a f ,(A2 ) a − f a − = f a − f for g ≥ MICHA(cid:32)L STUKOW (B2)
Using (A2), (A2 ), (A2 ) and (B4 ) we rewrite (B2).[ y ] → ( a a y − a − [ y ] → a a ) y = a ( a a y − a − ya a ) a , [ e − a − a − a − e − ] ← y = a a a f [ a a a ] ,y = e [ a a a ] ea a a f a a a ,y = [ e ] → a a a [ e ] ← a a a f a a a ,y = a [ ea e ] a a a a a f a a ,y = a a ea a a a a a f a a . In the above computations we introduced the notation which shouldhelp the reader to follow our transformations. The underlined partsindicate expressions which will be reduced, and parts with small arrowsindicate expressions which will be moved to the left/right.As a conjugation of (B2) we can take( a a y − a − ya a ) = y − a ( a a y − a − ya a ) a y − . By a straightforward computation this gives y − = a a f a a a a a a ea a , which together with (B2 ) gives (B2 ).Observe that (B2 ) together with (A1) and (A1 ) imply that we canreplace (B3) for i ≥ ).Observe also that (B2 ), (A2), (B4 ) and (B4 ) imply that (B3) for i = 1 is superfluous.We will now show that (D1) is superfluous – we will need here (A2 ),hence we add this relation to the statement. Using (B2 ) we substitutefor y . f = ([ a − ] ← a − f − a − a − a − a − a − a − e − a − a − )[ e ] ← ( a a e [ a ] → a a a a a f a [ a ] → ) ,f = ( a − f − a − a − a − a − a − e − a − )( a ea a a a a f a ) f . (B1) If we use (B1) in the form( a a a a ya − a − a − a − ) y − = y − ( a a a a ya − a − a − a − ) , after rewriting we get (B1). Conjugating this relation by y gives(B1 ) y ( a a a a [ y ] ← a − a − a − a − )[ y − ] → = ( a a a a ya − a − a − a − )[ y − ] ← ,y ( f − a a − f − a − a − a − a − ) = ( a a a a ea a − e ) y . FINITE PRESENTATION FOR THE TWIST SUBGROUP. . . 9
Now we will show that this relation is superfluous – it is a consequenceof relations (A1), (A2), (A2 )–(A2 ), (B1), (B2 ), (B2 ), (B4 ), (B4 ).We substitute for y using (B2 ) and (B2 ).( a a ea a a a a a f a a )([ f − ] ← a a − f − a − a − a − a − ) == ( a a a a ea a − [ e ] → )( a − a − e − a − a − a − a − a − a − f − a − a − ) , ( ea a [ a ] ← [ a ] → a f a a )( a [ a − ] ← f − a − a − ) == ( a a e [ a ] → a − )( a − a − e − a − [ a − ] ← a − a − [ a − ] → f − ) , ([ e ] ← a a f a a )( f a f − a − a − )[ a ] ← = [ a − ] → ( a a ea − )( e − a − a − e − a − a − [ f − ] → ) , ( a a f a a )( a [ f − ] → [ a − ] ← a − f ) = ( e − a [ a ] → [ e ] ← a − )( a − a − e − a − a − ) ,a a f a a − [ a ] → f [ a ] → = [ a − ] ← e − [ a − ] ← a a − e − a − a − ,a a a a f a a − f = e − a a − e − a − a − a − a − . What we get is (B1). (B6)
If we rewrite (B6) we get (B6 ), and (B6) conjugated by y gives(B6 ). (B7) If we use (B7) in the form a a a a a a a a ya − a − a − a − a − a − a − a − by − == ba a a a a a a a ya − a − a − a − a − a − a − a − y − , after rewriting we get (B7 ). By conjugating this relation by y − , takinginverses of both sides and using (D1), we get([ a a a a a a a a ] ← f a a − f a − a − a − a − )[ y − ] ← cy == b ( a a a a a a a a [ f a a − f a − a − a − a − ] → ) ,y − ( ea a − ea − a − a − a − ) c ( a a a a e − a a − e − ) y == ( a − a − a − a − a − a − a − a − ) b ( a a a a a a a a ) . This together with (B7 ) gives (B7 ). For further reference observethat using (B1) the above relation can be also rewritten as(B7 ) ( a a a a e − a a − e − )( a − a − a − a − a − a − a − a − ) y − cy == b ( a a a a e − a a − e − )( a − a − a − a − a − a − a − a − ) . Observe that we can use (B7 ) and (B6 ) as definitions of c . It isstraightforward to check that the first of these relations imply (A3 )and (A3 ) for i = 1. The second one imply (A3 ) for i = 3 and i ≥ (B8) If we rewrite (B8) we get (B8 ) and (B8) conjugated by y − gives(B8 ). Further reductions.
For any 3 ≤ k ≤ g − z k = ( a k − a k a k − a k − · · · a a e − a a − e − )( a − a − · · · a − k − a − k − a − k a − k − ) . Geometrically z k is the product of crosscap slides yY ± µ k ,α k , where µ k and α k are circles indicated in Figure 3 (see Section 4 of [15]), hence on theleft of µ k , conjugation by z k has the same effect as conjugation by y .More precisely,(D2) z k a z − k = a − (D3) z k a z − k = e − for k ≥ z k a i z − k = a i for 3 ≤ i ≤ k − z k bz − k = c for k ≥ z k y z − k = y ,(D7) z k f z − k = a − for k ≥ z k ez − k = y a − y − for k ≥ z k cz − k = y by − for k ≥ ), (A2 ), (A2 ). For (D5) we need additionally (A3), (A3 ) and(B7 ).Let us prove (D6) – we will use (A1), (A1 ), (B1), (B3) and (B1 )(hence we need all relations that we used to reduce (B1 )). z k y = ( a k − a k a k − a k − · · · a a e − a a − e − )( a − a − · · · a − k − a − k − a − k a − k − ) y == ( a k − a k · · · a a )[ e − a a − e − a − a − a − a − ] y ( a − a − · · · a − k a − k − ) == ( a k − a k · · · a a ) y [ e − a a − e − a − a − a − a − ]( a − a − · · · a − k a − k − ) = y z k . Now we will prove (D7) – we will use (A1), (A2), (A1 ), (A2 ), (B1). z k f = ( a k − a k · · · a a ) a a [ e − a a − e − a − a − a − a − ] a − a − ( a − a − · · · a − k a − k − ) f == ( a k − a k · · · a a ) a a [ a a a a f a a − f ] a − a − [ f ] ← ( a − a − · · · a − k a − k − ) == ( a k − a k · · · a a ) a − a a [ a a a a f a a − f ] a − a − ( a − a − · · · a − k a − k − ) == a − ( a k − a k · · · a a )[ e − a a − e − a − a − a − a − ]( a − a − · · · a − k a − k − ) = a − z k . Relation (D8) is a consequence of (D6), (D7) and (D1). Finally, (D9)is a consequence of (B7 ) and (D6) (hence we need (B7 )).Relations (D2)–(D9) imply that • (A4) is superfluous if g ≥ FINITE PRESENTATION FOR THE TWIST SUBGROUP. . . 11 • (A5) is superfluous if g ≥ • (A6) is superfluous if g ≥ • (B6 ) is superfluous if g ≥ • (B8 ) is superfluous if g ≥ g ≥
8, relations (A8a) and (A8b) for i < g − are con-sequences of relation (A8). Hence we can remove all these relationstogether with generators b , . . . , b g − and instead add the relation b i = z g − b i z − g − for i = g − , g − . This is exactly (A7c). (cid:3)
Theorem 3.2. If g ≥ is odd, then the group T ( N g, ) is isomorphic tothe quotient of the group T ( N g, ) with presentation given in Theorem3.1 obtained by adding a generator (cid:37) and relations (C1a) ( a a · · · a g − ) g = (cid:37) , (C1a) ( a − e − a · · · a g − ) g = y (cid:37) , (C2) a i (cid:37) = (cid:37)a i for i = 1 , , . . . , g − , (C2) (cid:37)e = f (cid:37) , (C5) (cid:37)y = y − (cid:37) , (C3) (cid:37) = 1 , (C4a) ( a a · · · a g − e − a · · · a g − ) g − = 1 .Moreover, relations (A1 ), (B2 ), (B4 ) are superfluous.If g ≥ is even, then the group T ( N g, ) is isomorphic to the quotientof the group T ( N g, ) with presentation given in Theorem 3.1 obtainedby adding a generator (cid:37) and relations (C1b) ( a a · · · a g − ) g = 1 , (C2 ) (cid:37)a = a − (cid:37) , (C2 ) (cid:37)a i = a i (cid:37) for i = 3 , . . . , g − , (C2 ) (cid:37)a = e − (cid:37) , (C5) (cid:37)y = y − (cid:37) , (C3) (cid:37) = 1 , (C4) ( (cid:37)a a · · · a g − ) g − = 1 .Moreover, relations (A1 ), ( A2 ), (A2 ) are superfluous.Proof. We follow the lines of the proof of Theorem 3.1, but as a startingpoint we now have Theorem 2.2. Moreover, it is convenient to add rela-tions (C2) and (C5), so in particular (C4a) and (C4b) are equivalent to(C4) (see Theorem 2.3). Generator (cid:37) yields two additional generatorsfor T ( N g, ), namely (cid:37), y(cid:37)y − if g is odd and (cid:37) = y(cid:37), (cid:37)y − if g is even. Suppose first that g is odd. Then (C5) and its conjugate by y − rewrite as y (cid:37) = y(cid:37)y − ,y (cid:37)y = (cid:37). The first relation implies that we can remove generator y(cid:37)y − – we willdo this silently from now on. The second one gives (C5).Relations (C1a), (C2), (C3) does not need rewriting, and if we con-jugate them by y we get respectively (C1a), (C2) (we use here (D1),hence also (A2 )) and relation equivalent to (C5).Relation (C4a) and its conjugate by y rewrite respectively as( f − a · · · a g − a a · · · a g − ) g − = 1 , ( a a · · · a g − e − a · · · a g − ) g − = 1 . The second relation is (C4a), and if we conjugate it by (cid:37) , by (C2) and(C2) we get the first one.Finally, observe that if we conjugate relations (A1 ), (B2 ), (B4 ) by (cid:37) we get respectively (A1 ), (B2 ), (B4 ).Now assume that g is even, hence (cid:37) = y(cid:37) ∈ T ( N g, ). Relation (C5)and its conjugate by y rewrite as y(cid:37) = (cid:37)y − ,y ( y(cid:37) ) = ( y(cid:37) ) y − . The first relation implies that we can remove generator (cid:37)y − – we willdo this silently from now on. The second one gives (C5).If we rewrite relation (C2) we get relations (C2 )–(C2 ).Relations (C1b), (C3) and (C4) rewrite respectively as (C1b), (C3)and (C4). Their conjugates by y ± are superfluous since, by (C2 )–(C2 ), they are the same as conjugates by (cid:37) .Finally, observe that if we conjugate relations (A1), (A2) by (cid:37) weget respectively (A1 ), (A2 )–(A2 ). (cid:3) Remark . Observe that relations (B2 ) and (C1a), (C4) allows toremove y and (cid:37), (cid:37) from the generating sets, hence the generating setsof the presentations given by Theorems 3.1 and 3.2 are really Dehntwists about nonseparating circles.4. Geometric interpretation
We devote this last section to the geometric interpretation of rela-tions obtained in Theorem 3.1.
FINITE PRESENTATION FOR THE TWIST SUBGROUP. . . 13
Relations (A1), (A3), (A9a), (A9b), (A1 ), (A1 ), (A3 ), (A3 ),(B3), (B4 ), (B4 ), (B7 ), (A9a), (A9b) are standard commutativityrelations between Dehn twists with disjoint supports.Relations (A2), (A4), (A2 )–(A2 ), (A4) are standard braid relationsbetween Dehn twists about circles intersecting in one point.Relations (A5), (A6), (A8), (A5), (A6), (A8a), (A8b) came fromMatsumoto [11] presentation of mapping class group of an orientablesurface. They have simple interpretation as relations between funda-mental elements of Artin groups – for details see [11] and [7].Relations (B7 ) and (A7c) are simple conjugation relations of theform t f ( α ) = f t α f − , where t α is the twist about a circle α .Relations (B6 ) and (B8 ) are conjugates (by y ± ) of (B6 ) and (B8 )respectively, and (B2 ) is equivalent to the conjugation of (B2 ), hencewe are left with four interesting relations: (B1), (B2 ), (B6 ) and (B8 ).Relation (B1) can be rewritten in a slightly more symmetric form( a ea ) a − ( a ea ) a ( a f a ) a − ( a f a ) a = 1 . This is a relation between five Dehn twists a , a , a , e, f illustrated inFigures 1 and 4.Relation (B2 ) can be rewritten as y = ( a ea ) ( a f a ) . This is a relation between five twists a , a , e, f, y illustrated in Figures1 and 4.Relation (B6 ) is a relation between four Dehn twists b, c, f (cid:48) = ( a a a ) f − ( a a a ) − , e (cid:48) = ( a a ) − e − ( a a ) , illustrated in Figures 1, 4 and 5 Figure 5.
Dehn twists e (cid:48) and f (cid:48) .Finally, relation (B8 ) is a relation between six Dehn twists c (cid:48) = ( a ea − a − ) c ( a ea − a − ) − , b (cid:48) = ( a a a a ) − b − ( a a a a ) ,a , a (cid:48) = ( a − ea a ) − a ( a − ea a ) , a , e. illustrated in Figures 1, 4 and 6. Figure 6.
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Institute of Mathematics, University of Gda´nsk, Wita Stwosza 57,80-952 Gda´nsk, Poland
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