A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties
aa r X i v : . [ m a t h . G T ] J un A CONJUGATION-FREE GEOMETRICPRESENTATION OF FUNDAMENTAL GROUPS OFARRANGEMENTS II: EXPANSION AND SOMEPROPERTIES
MEITAL ELIYAHU , DAVID GARBER AND MINA TEICHER Abstract. A conjugation-free geometric presentation of a funda-mental group is a presentation with the natural topological gener-ators x , . . . , x n and the cyclic relations: x i k x i k − · · · x i = x i k − · · · x i x i k = · · · = x i x i k · · · x i with no conjugations on the generators.We have already proved in [13] that if the graph of the arrange-ment is a disjoint union of cycles, then its fundamental group has aconjugation-free geometric presentation. In this paper, we extendthis property to arrangements whose graphs are a disjoint unionof cycle-tree graphs.Moreover, we study some properties of this type of presenta-tions for a fundamental group of a line arrangement’s complement.We show that these presentations satisfy a completeness propertyin the sense of Dehornoy, if the corresponding graph of the ar-rangement has no edges. The completeness property is a powerfulproperty which leads to many nice properties concerning the pre-sentation (such as the left-cancellativity of the associated monoidand yields some simple criterion for the solvability of the wordproblem in the group). Introduction
The fundamental group of the complement of plane curves is a veryimportant topological invariant, which can be also computed for linearrangements. We present here some applications of this invariant.Chisini [6], Kulikov [20, 21] and Kulikov-Teicher [22] have used thefundamental group of complements of branch curves of generic pro-jections in order to distinguish between connected components of themoduli space of smooth projective surfaces, see also [15]. Partially supported by the Israeli Ministry of Science and Technology.
Key words and phrases. conjugation-free presentation, fundamental group, com-plete presentation, complemented presentation
MSC2010:
Primary: 14H30; Secondary: 32S22, 57M05, 20M05, 20F05.
Moreover, the Zariski-Lefschetz hyperplane section theorem (see [24])states that: π ( CP N \ S ) ∼ = π ( H \ ( H ∩ S )) , where S is a hypersurface and H is a generic 2-plane. Since H ∩ S isa plane curve, the fundamental groups of complements of curves canbe used also for computing the fundamental groups of complements ofhypersurfaces in CP N .A different need for computations of fundamental groups is to obtainmore examples of Zariski pairs [31, 32]. A pair of plane curves is called a Zariski pair if they have the same combinatorics (to be exact: there isa degree-preserving bijection between the set of irreducible componentsof the two curves C , C , and there exist regular neighbourhoods of thecurves T ( C ) , T ( C ) such that the pairs ( T ( C ) , C ) , ( T ( C ) , C ) arehomeomorphic and the homeomorphism respects the bijection above[3]), but their complements in P are not homeomorphic. For a survey,see [5].It is also interesting to explore new finite non-abelian groups whichserve as fundamental groups of complements of plane curves in general;see for example [1, 2, 12, 31].An affine line arrangement in C is a union of copies of C in C .Such an arrangement is called real if the defining equations of all itslines can be written with real coefficients, and complex otherwise. Notethat the intersection of a real arrangement with the natural copy of R in C is an arrangement of lines in the real plane, called the real part of the arrangement.Similarly, a projective line arrangement in CP is a union of copiesof CP in CP . Note that the realization of the MacLane configuration[23] is an example of a complex arrangement; see also [4, 28].For real and complex line arrangements L , Fan [14] defined a graph G ( L ) which is associated to its multiple points (i.e., points where morethan two lines are intersected). We give here its version for real arrange-ments: Given a real line arrangement L , the graph G ( L ) of multiplepoints lies on the real part of L . It consists of the multiple points of L , with the segments between the multiple points on lines which haveat least two multiple points. Note that if the arrangement consists ofthree multiple points on the same line, then G ( L ) has three vertices onthe same line (see Figure 1(a)). If two such lines happen to intersectin a simple point (i.e., a point where exactly two lines are intersected),it is ignored (i.e., the lines do not meet in the graph). See another ex-ample in Figure 1(b) (note that Fan’s definition gives a graph differentfrom the graph defined in [18, 29]). ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 3 (a) (b)
Figure 1.
Examples for G ( L )In [13] we introduce the notion of a conjugation-free geometric pre-sentation of the fundamental group of an arrangement: Definition 1.1.
Let G be a fundamental group of the affine or projec-tive complements of some line arrangement with n lines. We say that G has a conjugation-free geometric presentation if G has a presentationwith the following properties: • In the affine case, the generators { x , . . . , x n } are the meridiansof lines at some far side of the arrangement, and therefore thenumber of generators is equal to n . • In the projective case, the generators are the meridians of linesat some far side of the arrangement except for one, and there-fore the number of generators is equal to n − . • In both cases, the relations are of the following type: x i k x i k − · · · x i = x i k − · · · x i x i k = · · · = x i x i k · · · x i , where { i , i , . . . , i k } ⊆ { , . . . , m } is an increasing subsequenceof indices, where m = n in the affine case and m = n − in the projective case. Note that for k = 2 , we get the usualcommutator. Note that in usual geometric presentations of the fundamental group,most of the relations have conjugations.The importance of this family of arrangements is that the funda-mental group can be read directly from the arrangement or equivalentlyfrom its incidence lattice (where the incidence lattice of an arrangementis the partially-ordered set of non-empty intersections of the lines, or-dered by inclusion, see [27]) without any computation. Hence, for thisfamily of arrangements, the incidence lattice determines the fundamen-tal group of the complement (this is based on Cordovil [7], too).
ELIYAHU, GARBER, TEICHER
We start with the easy fact that there exist arrangements whose fun-damental groups have no conjugation-free geometric presentation: Thefundamental group of the Ceva arrangement (also known as the braidarrangement , appears in Figure 2) has no conjugation-free geometricpresentation (see [13]).
Figure 2.
Ceva arrangementNote also that if the fundamental groups of two arrangements L , L have conjugation-free geometric presentations and the arrangementsintersect transversally, then the fundamental group of L ∪ L has aconjugation-free geometric presentation, too. This is due to the impor-tant result of Oka and Sakamoto [26]: Theorem 1.2. (Oka-Sakamoto)
Let C and C be algebraic planecurves in C . Assume that the intersection C ∩ C consists of distinct d · d points, where d i ( i = 1 , are the respective degrees of C and C . Then: π ( C − ( C ∪ C )) ∼ = π ( C − C ) ⊕ π ( C − C )The main result of [13] is: Proposition 1.3.
The fundamental groups of following family of ar-rangements have a conjugation-free geometric presentation: a real ar-rangement L , where G ( L ) is a disjoint union of cycles of any length,and the multiplicities of the multiple points are arbitrary. In this paper, we continue the investigation of the family of arrange-ments whose fundamental groups have conjugation-free geometric pre-sentations in two directions. First, we extend this property to realarrangements whose graphs are a disjoint union of cycle-tree graphs,where an example for a cycle-tree graph is presented in Figure 3 (seeDefinition 2.4 below).
ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 5
Figure 3.
An example of a cycle-tree graphIn the second direction, we study some properties of this type of pre-sentations for a fundamental group of a line arrangement’s complement.We prove:
Proposition 1.4.
Let L be a real arrangement whose fundamentalgroup has a conjugation-free geometric presentation and its graph G ( L ) has no edges. Then the presentation of the corresponding monoid iscomplete (and complemented). The completeness property is a powerful property which leads tomany nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterionfor the solvability of the word problem in the group and for Garsidegroups).The paper is organized as follows. In Section 2, we prove that ar-rangements whose graphs are a disjoint union of cycle-tree graphs havea conjugation-free geometric presentation of the fundamental group ofthe complement. In Section 3, we prove that conjugation-free geomet-ric presentations are complemented presentations. Section 4 deals withcomplete presentations, and includes the proof of Proposition 1.4.2.
Adding a line through a single point preserves theconjugation-free geometric presentation
We start with the following obvious observation, which is based onthe Oka-Sakamoto decomposition theorem (see Theorem 1.2 above):
Observation 2.1.
Let L be an arrangement whose fundamental grouphas a conjugation-free geometric presentation. Let L be a line whichintersects L transversally. Then L ∪ L is also an arrangement whosefundamental group has a conjugation-free geometric presentation. In this section, we prove the following proposition, which is the nextstep:
ELIYAHU, GARBER, TEICHER
Proposition 2.2.
Let L be a real arrangement whose affine fundamen-tal group has a conjugation-free geometric presentation. Let L be a linenot in L , which passes through one intersection point P of L . Then L ∪ L is also an arrangement whose affine fundamental group has aconjugation-free geometric presentation.Proof. We can assume that the point P is the leftmost and lowestpoint of the arrangement L and all the intersection points of the line L (except for P ) are to the left of all the intersection points of thearrangement L (except for P ). We can also assume that the highestline in L (with respect to the global numeration of the lines) passesthrough P . See Figure 4 for an illustration, where the arrangement L is in the dashed rectangle. L P
Figure 4.
An illustration of the real part of
L ∪ L The above assumption is due to the following: First, one can rotatea line that participates in only one multiple point as long as it doesnot unite with a different line (by Results 4.8 and 4.13 of [17]). Sec-ond, moving a line that participates in only one multiple point over adifferent line (see Figure 5) is permitted in the case of a triangle dueto a result of Fan [14] that the family of configurations with 6 linesand three triple points is connected by a finite sequence of smooth eq-uisingular deformations. Moreover, by Theorem 4.11 of [17], one canassume that the point P is the leftmost point of the arrangement L .Let n be the number of lines in L and let m be the multiplicity of P in L . So, the list of Lefschetz pairs of the arrangement L is([ a , b ] , [ a , b ] , . . . , [ a q − , b q − ] , [1 , m ]) , where the Lefschetz pair [1 , m ] corresponds to the point P (for the the-ory used here for computing the fundamental group of the complementsof arrangements, see [13, 16, 19, 25]). Since we have that π ( C − L )has a conjugation-free geometric presentation, then we know that all ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 7
33 33 3 3
Figure 5.
Moving a line that participates in only onemultiple point over a different linethe conjugations in the relations induced by the van Kampen Theorem[19] (see also [13]) can be simplified.Now, let us deal with the arrangement
L ∪ L . By our assumptions,its list of Lefschetz pairs is (we write in small brackets the name of thecorresponding point):([ a + 1 , b + 1] ( p ) , [ a + 1 , b + 1] ( p ) , . . . , [ a q − + 1 , b q − + 1] ( p q − ) , [1 , m +1] ( p q ) , [ m +1 , m +2] ( p q +1 ) , [ m +2 , m +3] ( p q +2 ) , . . . , [ n, n +1] ( p q +( n − m ) ) ) . We start with the relations induced from intersection points on theline L . We first choose a set of n +1 generators of the fundamental groupof its complement corresponding to its lines, namely { x , . . . , x n +1 } .By the Moishezon-Teicher algorithm [16, 25] (see also [13]), we nowcompute the skeletons corresponding to the points on the line L (i.e.the points p j , where q ≤ j ≤ q + n − m ). Note that: ∆ h a + 1 , b + 1 i ∆ h a + 1 , b + 1 i · · · ∆ h a q − + 1 , b q − + 1 i = ∆ h , n + 1 i ∆ − h , m + 1 i , since given an arrangement, the multiplication of all the halftwistsbased on its Lefschetz pairs is equivalent to a unique halftwist of allthe lines. By this observation, we get the skeletons in Figure 6. p : q p : q+j 1 n−m+2−j n+11 n+1n−m+2 n−m+3 Figure 6.
The skeletons of the points p q + j where0 ≤ j ≤ n − m Hence, we get the following relations:For the point p q : x n +1 x n · · · x n − m +2 x = x x n +1 x n · · · x n − m +2 == · · · = x n · · · x n − m +2 x x n +1 ELIYAHU, GARBER, TEICHER
For the points p j , where q + 1 ≤ j ≤ q + n − m : x n − m +2 − j x = x x n − m +2 − j These relations are obviously without conjugations.Now, we move to the relations induced from points appearing inthe original arrangement L . The only change in the level of the Lef-schetz pairs is an addition of one index in all the pairs, due to the line L . Therefore, the induced braid monodromy and the relations will bechanged by adding 1 to every index, i.e., if we have a relation whichinvolves the generators x i , . . . , x i k , then after adding the line, we havethe same relation but with generators x i +1 , . . . , x i k +1 , respectively.Now, we know that the fundamental group of L has a conjugation-free geometric presentation; hence we have that by a simplificationprocess, one can reach a presentation without conjugations. If we im-itate the simplification process of the presentation of the fundamentalgroup of L for the presentation of the fundamental group of L ∪ L , thecases in which we need to use the relations induced from the point P are the relations that have been simplified by using the relations in-duced from P before adding the line. As above, the original relationsinduced from P are: R p : x n x n − · · · x n − m +1 = x n − · · · x n − m +1 x n == · · · = x n − m +1 x n · · · x n − m +2 , while the new ones are:˜ R p : x n +1 x n · · · x n − m +2 x = x x n +1 x n · · · x n − m +2 == · · · = x n · · · x n − m +2 x x n +1 . We can divide the relations induced from L before adding the line L into two subsets:(1) Relations that during the simplification process contain the sub-word x − n − m +2 · · · x − n − x n x n − · · · x n − m +2 .(2) Relations that do not contain the above subword during itssimplification process.For the second subset, the simplification process will be identical be-fore adding the line L and after it, since all the other relations inducedby L have not been changed by adding the line L (except for adding 1to the indices).For the first subset, let us denote the relation by R . Except forapplying the relations induced from P , the rest of simplification processis identical to the one before adding the line (again, except for adding1 to the indices). The only change is in the step of applying R p . In this ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 9 step, before adding the line L , the generator x has not been involvedin R p , but after adding the line L , it appears in ˜ R p . Hence, for applying˜ R p , we have to conjugate relation R by x , and using the commutativerelations which x is involved in, we can diffuse x into the relation R ,so we can use the relation ˜ R p instead of R p .Hence, we can simplify all the conjugations in all the relations, sowe have a conjugation-free geometric presentation, as needed. (cid:3) Remark 2.3.
Note that adding a line which closes a cycle in L mightnot preserve the conjugation-free geometric presentation property. Forexample, adding a line to an arrangement of 5 lines which creates theCeva arrangement (see Figure 2) is not an action which preserves theconjugation-free geometric presentation property. Hence, we can extend the family of arrangements whose fundamentalgroups have a conjugation-free geometric presentation. We start withthe following definition:
Definition 2.4. A cycle-tree graph is a graph which consists of a cycle,where each vertex of the cycle can be a root of a tree, see Figure 7. Itis possible that there exist some vertices also in the middle of an edgeof the cycle or the trees. Figure 7.
An example of a cycle-tree graph
Corollary 2.5.
Let L be a real line arrangement whose graph is adisjoint union of cycle-tree graphs. Then the fundamental group of L has a conjugation-free geometric presentation.Proof. We start by proving that a real arrangement whose graph isa cycle-tree graph has a fundamental group which has a conjugation-free geometric presentation. We already have from [13] that a realarrangement whose graph is a cycle has a fundamental group which hasa conjugation-free geometric presentation. By Proposition 2.2, addinga line which is either transversal to an arrangement or passes throughone intersection point, preserves the property that the fundamental group has a conjugation-free geometric presentation. One can easilyconstruct an arrangement whose graph is a cycle-tree graph from anarrangement whose graph is a cycle by inductively adding a line whichis either transversal to the arrangement or passes through one of itsintersection points. Hence, we get that an arrangement whose graph isa cycle-tree graph has a fundamental group which has a conjugation-free geometric presentation.In the next step, using the theorem of Oka and Sakamoto [26] (seeTheorem 1.2 above), we can generalize the result from the case of onecycle-tree graph to the case of a disjoint union of cycle-tree graphs. (cid:3) Complemented presentations A semigroup presentation ( S , R ) consists of a nonempty set S anda family of pairs of nonempty words R in the alphabet S . The corre-sponding monoid ( S , R ) is hS|Ri + ∼ = ( S ∗ / ≡ + R ).Dehornoy [8] has defined the notion of a complemented presentation of a semigroup: Definition 3.1.
A semigroup presentation ( S , R ) is called comple-mented if, for each s ∈ S , there is no relation s . . . = s . . . in R and,for s, s ′ ∈ S , there is at most one relation s . . . = s ′ . . . in R . Our type of presentations satisfies this property as follows:
Lemma 3.2.
A conjugation-free geometric presentation is a comple-mented presentation.Proof.
Any pair of lines intersect exactly once, hence their correspond-ing generators appear as prefixes in exactly one relation. Since thereare no conjugations, this is their unique appearance as a pair of pre-fixes. (cid:3)
Remark 3.3. (1)
This property does not hold for presentations of fundamentalgroups in general (due to the conjugations in the relations). (2)
This property does not hold in the homogeneous minimal pre-sentations introduced by Yoshinaga [30] . Complete presentations
In this section, we will study which cases of conjugation-free geomet-ric presentations are also complete in the sense of Dehornoy [10]. Thecompleteness property is a very important and powerful property. In
ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 11
Section 4.1, we supply some background on this property and some ofits applications. In Section 4.2, we present our results in this direction.4.1.
Background on complete presentations.
We follow the sur-vey of Dehornoy [11]. We start by defining the notion of a word revers-ing : Definition 4.1.
For a semigroup presentation ( S , R ) and w, w ′ ∈ ( S ∪ S − ) ∗ , w reverses to w ′ in one step , denoted by w y R w ′ ,if there exist a relation sv ′ = s ′ v of R and u, u ′ satisfying: w = us − s ′ u ′ and w ′ = uv ′ v − u ′ . We say that w reverses to w ′ in k steps , denoted by w y kR w ′ , if thereexist words w , . . . , w k satisfying w = w, w k = w ′ and w i y R w i +1 for each i . The sequence ( w , . . . , w k ) is called an R -reversing sequencefrom w to w ′ .We write w y w ′ , if w y k R w ′ holds for some k ∈ N . Definition 4.2.
A semigroup presentation ( S , R ) is called complete if, for all words w, w ′ ∈ S ∗ : w ≡ + R w ′ ⇒ w − w ′ y R ε. where ε is the empty word. In the next definition, we define the cube condition , which is a usefultool for verifing the completeness property.
Definition 4.3.
Let ( S , R ) be a semigroup presentation, and u, u ′ , u ′′ ∈ S ∗ . We say that ( S , R ) satisfies the cube condition for( u, u ′ , u ′′ ) if: u − u ′′ u ′′− u ′ y R v ′ v − ⇒ ( uv ′ ) − ( vu ′ ) y R ε. For X ⊆ S ∗ , we say that ( S , R ) satisfies the cube condition on X if it satisfies the cube condition for every triple ( u, u ′ , u ′′ ) where u, u ′ , u ′′ ∈ X . u u’’ u’ vv’u’’ uv’ u’ v Figure 8.
An illustration of the cube condition
Definition 4.4.
A semigroup presentation ( S , R ) is said to be homo-geneous if there exists an ≡ + R -invariant mapping λ : S ∗ → N satisfying,for s ∈ S and w ∈ S ∗ , λ ( sw ) > λ ( w ) . A typical case of a homogeneous presentation is where all relationsin R preserve the length of words, i.e., they have the form v ′ = v where v ′ and v have the same length.Dehornoy [10] has proved the following result: Proposition 4.5.
Assume that ( S , R ) is a homogeneous semigrouppresentation. Then ( S , R ) is complete if and only if it satisfies thecube condition on S . The next definition is needed for introducing an operation used inan equivalent condition for the cube condition:
Definition 4.6.
For a complemented semigroup presentation ( S , R ) and w, w ′ ∈ S ∗ , the R -complement of w ′ in w , denoted w \ w ′ , (“ w under w ′ ”), is the unique word v ′ ∈ S ∗ such that w − w ′ reverses to v ′ v − for some v ∈ S ∗ , if such a word exists. Dehornoy [10] has proved that the cube condition is equivalent tosome expression involving the complement operation:
Proposition 4.7.
Assume that ( S , R ) is a complemented semigrouppresentation. Then, for all words u, u ′ , u ′′ ∈ S ∗ , the following are equiv-alent: (1) ( S , R ) satisfies the cube condition on { u, u ′ , u ′′ } . (2) either ( u \ u ′ ) \ ( u \ u ′′ ) and ( u ′ \ u ) \ ( u ′ \ u ′′ ) are R -equivalent orthey are not defined, and the same holds for all permutations of u, u ′ , u ′′ . Here, we survey some important consequences and applications aris-ing from the completeness property.
Proposition 4.8 ([10], Proposition 6.1) . Every monoid that admitsa complete complemented presentation is left-cancellative (i.e., xy = xz ⇒ y = z ). Proposition 4.9 ([10], Proposition 6.10) . Assume that ( S , R ) is acomplete semigroup presentation. If ( S , R ) is complemented, then themonoid hS|Ri + admits least common multiples. ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 13
Completeness of conjugation-free geometric presentations.
In this section, we prove that a conjugation-free geometric presentationis complete if its corresponding graph has no edges:
Proposition 4.10.
Let L be a real arrangement whose fundamentalgroup has a conjugation-free geometric presentation and its graph G ( L ) has no edges. Then the presentation of the corresponding monoid iscomplete (and complemented).Proof. It is obvious that the conjugation-free geometric presentationsare homogeneous (since all the words in the same relation are of thesame length). Hence, we prove this proposition by verifying the equiv-alent version of the cube condition (for any triple ( u, u ′ , u ′′ ) ∈ ( S ∗ ) ,the words ( u \ u ′ ) \ ( u \ u ′′ ) and ( u ′ \ u ) \ ( u ′ \ u ′′ ) are R -equivalent) case-by-case.Case 1: The three generators correspond to three lines ℓ i , ℓ j , ℓ k inter-secting in three simple points, see Figure 9. i jk Figure 9.
Case 1In this case, the relations induced by the three simple pointsare: [ x i , x j ] = [ x i , x k ] = [ x j , x k ] = e , where x i , x j , x k are thegenerators of the lines ℓ i , ℓ j , ℓ k respectively. So, we have:( x i \ x j ) \ ( x i \ x k ) = x k = ( x j \ x i ) \ ( x j \ x k ) , which are indeed R -equivalent.By symmetry, this holds to any permutation of x i , x j , x k asneeded.Case 2: The three generators correspond to three lines ℓ i , ℓ j , ℓ k passingthrough the same multiple point, see Figure 10. For this case,we have two subcases: in the first case, the corresponding linesappear consecutively in the intersection point. In the secondcase, the corresponding lines appear separately in the intersec-tion point.Case 2a: The lines appear consecutively in the intersection point:
Without loss of generality, we can assume that the multi-ple point has multiplicity 4, see Figure 10(a). Hence, the ji k ji1 k2 3(b)(a)
Figure 10.
Case 2relations induced by this multiple point are: xx k x j x i = x k x j x i x = x j x i xx k = x i xx k x j , where x i , x j , x k , x are the generators of the lines ℓ i , ℓ j , ℓ k , ℓ respectively. Hence, we have:( x i \ x j ) \ ( x i \ x k ) = e = ( x j \ x i ) \ ( x j \ x k ) , which are indeed R -equivalent.Any other permutation of x i , x j , x k yields e in both sidesof the condition, so the condition is satisfied for any per-mutation.Case 2b: The lines do not appear consecutively in the intersectionpoint:
Without loss of generality, we can assume that themultiple point has multiplicity 6, see Figure 10(b). Hence,the relations induced by this multiple point are: zx k yx j xx i = x k yx j xx i z = yx j xx i zx k = x j xx i zx k y == xx i zx k yx j = x i zx k yx j x, where x i , x j , x k , x, y, z are the generators of the lines ℓ i , ℓ j , ℓ k , ℓ , ℓ , ℓ respectively. Hence, we have:( x i \ x j ) \ ( x i \ x k ) = e = ( x j \ x i ) \ ( x j \ x k ) , which are indeed R -equivalent.Any other permutation of x i , x j , x k yields e in both sidesof the condition, so the condition is satisfied for any per-mutation.Hence, we have verified the equivalent version of the cube condition(( u \ u ′ ) \ ( u \ u ′′ ) and ( u ′ \ u ) \ ( u ′ \ u ′′ ) are R -equivalent) for any triple ofgenerators u, u ′ , u ′′ in any case that the graph has no edges, so we aredone. (cid:3) Hence, we have the following corollary:
ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 15
Corollary 4.11.
Let L be a real arrangement whose fundamental grouphas a conjugation-free geometric presentation and its graph G ( L ) hasno edges. Then, the corresponding monoid is cancellative and has leastcommon multiples. Remark 4.12.
The condition that the graph has no edges is essential,since if we take a line arrangement whose graph contains an edge, andits fundamental group has a conjugation-free geometric presentation,we can find a triple of generators for which the cube condition is notsatisfied anymore.
Acknowledgments
We would like to thank an anonymous referee of our previous paper[13] for pointing us out the possible connection between the conjugation-free geometric presentations and Dehornoy’s complete presentations.We owe special thanks to Patrick Dehornoy for many discussions.We wish to thank Tadashi Ishibe for pointing us a wrong consequencefrom the published version of this paper, and consequently it weakensour result concerning complete presentations.
References [1] E. Artal-Bartolo,
Fundamental group of class of rational cuspidal curves ,Manuscripta Math. , 273–281 (1997).[2] E. Artal-Bartolo, A curve of degree five with non-abelian fundamental group ,Topology Appl. , 13–29 (1997).[3] E. Artal-Bartolo and J. Carmona-Ruber, Zariski pairs, fundamental groupsand Alexander polynomials , J. Math. Soc. Japan (3), 521–543 (1998).[4] E. Artal-Bartolo, J. Carmona-Ruber, J.I. Cogolludo and M.A. Marco, Invari-ants of combinatorial line arrangements and Rybnikov’s example , in: Singular-ity theory and its applications, 1–34, Adv. Stud. Pure Math. , Math. Soc.Japan, Tokyo, 2006.[5] E. Artal-Bartolo, J.I. Cogolludo and H. Tokunaga, A survey on Zariski pairs ,in:
Algebraic geometry in east Asia, Hanoi 2005 , Adv. Stud. Pure Math. ,Math. Soc. Japan, Tokyo, 1–100 (2008).[6] O. Chisini, Sulla identit`a birazionale delle funzioni algebriche di due variabilidotate di una medesima curva di diramazione , Rend. Ist. Lombardo , 339–356 (1944).[7] R. Cordovil, The fundamental group of the complement of the complexificationof a real arrangement of hyperplanes , Adv. Appl. Math. , 481–498 (1998).[8] P. Dehornoy, Groups with a complemented presentation , J. Pure Appl. Alg. , 115–137 (1997).[9] P. Dehornoy,
Groupes de Garside , Ann. Scient. Ec. Norm. Sup. , 267–306(2002).[10] P. Dehornoy, Complete positive group presentations , J. Alg. , 156–197(2003). [11] P. Dehornoy,
The subword reversing method , preprint (2009) [Available online:http://arxiv.org/abs/0912.4272].[12] A.I. Degtyarev,
Quintics in CP with nonabelian fundamental group , Algebrai Analiz (5), 130–151 (1999) [Russian]; English translation: St. PetersburgMath. J. (5), 809–826 (2000).[13] M. Eliyahu, D. Garber and M. Teicher, A conjugation-free geometric presen-tation of fundamental groups of arrangements , Manuscripta Math. (1–2),247–271 (2010).[14] K.M. Fan,
Direct product of free groups as the fundamental group of the com-plement of a union of lines , Michigan Math. J. (2), 283–291 (1997).[15] M. Friedman and M. Teicher, On non fundamental group equivalent surfaces ,Alg. Geom. Topo. , 397–433 (2008).[16] D. Garber and M. Teicher, The fundamental group’s structure of the comple-ment of some configurations of real line arrangements , Complex Analysis andAlgebraic Geometry, edited by T. Peternell and F.-O. Schreyer, de Gruyter,173-223 (2000).[17] D. Garber, M. Teicher and U. Vishne,
Classes of wiring diagrams and theirinvariants , J. Knot Theory Ramifications (8), 1165–1191 (2002).[18] T. Jiang and S.S.-T. Yau, Diffeomorphic types of the complements of arrange-ments of hyperplanes , Compositio Math. (2), 133–155 (1994).[19] E.R. van Kampen, On the fundamental group of an algebraic curve , Amer. J.Math. , 255–260 (1933).[20] V.S. Kulikov, On Chisini’s conjecture , Izv. Ross. Akad. Nauk Ser. Mat. (6),83–116 (1999) [Russian]; English translation:
Izv. Math. , 1139–1170 (1999).[21] V.S. Kulikov, On Chisini’s conjecture II , Izv. Ross. Akad. Nauk Ser. Mat. (5), 63–76 (2008) [Russian]; English translation:
Izv. Math. (5), 901–913(2008).[22] V.S. Kulikov and M. Teicher, Braid monodromy factorizations and diffeomor-phism types , Izv. Ross. Akad. Nauk Ser. Mat. (2), 89–120 (2000) [Russian]; English translation:
Izv. Math. (2), 311–341 (2000).[23] S. MacLane, Some interpretations of abstract linear independence in terms ofprojective geometry , Amer. J. Math. , 236–241 (1936).[24] J. Milnor, Morse Theory , Ann. Math. Stud. , Princeton University Press,Princeton, NJ (1963).[25] B. Moishezon and M. Teicher, Braid group technique in complex geometry I:Line arrangements in CP , Contemp. Math. , 425–555 (1988).[26] M. Oka and K. Sakamoto, Product theorem of the fundamental group of areducible curve , J. Math. Soc. Japan (4), 599–602 (1978).[27] P. Orlik and H. Terao, Arrangements of hyperplanes , Grundlehren der Math-ematischen Wissenschaften , Springer-Verlag, Berlin, 1992.[28] G. Rybnikov,
On the fundamental group of the complement of a complex hy-perplane arrangement , DIMACS Tech. Report , 33–50 (1994).[29] S. Wang and S.S.-T. Yau,
Rigidity of differentiable structure for new class ofline arrangements , Comm. Anal. Geom. (5), 1057–1075 (2005).[30] M. Yoshinaga, Hyperplane arrangements and Lefschetz’s hyperplane sectiontheorem , Kodai Math. J. , 157–194 (2007).[31] O. Zariski, On the problem of existence of algebraic functions of two variablespossessing a given branch curve , Amer. J. Math. , 305–328 (1929). ONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS II 17 [32] O. Zariski,
On the Poincar´e group of rational plane curves , Amer. J. Math. , 607–619 (1936). Meital Eliyahu, Department of Mathematics, Bar-Ilan University,52900 Ramat-Gan, Israel
E-mail address : [email protected] David Garber, Department of Applied Mathematics, Faculty of Sci-ences, Holon Institute of Technology, 52 Golomb st., PO Box 305,58102 Holon, Israel
E-mail address : [email protected] Mina Teicher, Department of Mathematics, Bar-Ilan University,52900 Ramat-Gan, Israel
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