## CONJUGATION-FREE GEOMETRIC PRESENTATIONS OF FUNDAMENTAL GROUPS OF ARRANGEMENTS (2008)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Eliyahu08conjugation-freegeometric,

author = {Meital Eliyahu and David Garber and Mina Teicher},

title = {CONJUGATION-FREE GEOMETRIC PRESENTATIONS OF FUNDAMENTAL GROUPS OF ARRANGEMENTS},

year = {2008}

}

### OpenURL

### Abstract

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of line arrangements’ complements, and we show that the fundamental group of following family of arrangements have a conjugation-free geometric presentation: An arrangement L, whose graph of multiple points is a unique cycle of length n, and the multiplicities of the multiple points are arbitrary. We also compute the exact structure (by means of a semi-direct product of groups) of the arrangement which consists of a cycle of length 3, where all the multiple points are of multiplicity 3.

### Citations

70 | Braid group techniques in complex geometry III: Projective degeneration of V3 - Moishezon, Teicher - 1994 |

62 |
On the fundamental group of an algebraic curve
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(Show Context)
Citation Context ...g the relations (where ai are words in a group): anan−1 · · ·a1 = an−1 · · ·a1an = · · · = a1an · · ·a2,8 ELIYAHU, GARBER, TEICHER we will write: [a1, a2, · · · , an]. By the van Kampen theorem (see =-=[10]-=-), we get the following presentation of the fundamental group’ of the line arrangement’s complement: Generators: {x1, x2, . . ., x3n} Relations: (1) [xi, x −1 n+1 · · ·x −1 j−1 xjxj−1 · · ·xn+1] = e, ... |

61 |
On the problem of existence of algebraic functions of two variables possessing a given branch curve
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(Show Context)
Citation Context ...also for computing the fundamental group of complements of hypersurfaces in CP N . A different direction for the need of fundamental groups’ computations is for getting more examples of Zariski pairs =-=[18, 19]-=-. A pair of plane curves is called a Zariski pair if they have the same combinatorics (i.e. the same singular points and the same arrangement of irreducible components), but their complements are not ... |

27 | Homology of iterated semidirect products of free groups
- Cohen, Suciu
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(Show Context)
Citation Context ...·x2n+1x1x3n = · · · = x1x3n · · ·x2n+110 ELIYAHU, GARBER, TEICHER as needed, and hence we are done. □ 2.2. The structure of the fundamental group of the simplest case of this family. Cohen and Suciu =-=[2]-=- give the following presentation of F3 ⋊α3 F2 ⋊α2 F1, which is known to be the fundamental group of the complement of the Ceva arrangement (see Figure 1): F1 = 〈u〉, F2 = 〈t, s〉, F3 = 〈x, y, z〉 The act... |

26 | Braid monodromy factorization and diffeomorphism types, Izvestiya: Mathematics 64:2
- S, Teicher
- 2000
(Show Context)
Citation Context ... 3. 1. Introduction The fundamental group of the complement is a very important topological invariant of arrangements of lines. This invariant was used by Chisini [1], Kulikov [12] and KulikovTeicher =-=[13]-=- in order to distinguish between connected components of the moduli space of surfaces of general type. Moreover, the Zariski-Lefschetz hyperplane section theorem (see [15]) showed that π1(CP N \ S) ∼ ... |

23 | On Chisini’s conjecture
- Kulikov
- 1999
(Show Context)
Citation Context ...ints are of multiplicity 3. 1. Introduction The fundamental group of the complement is a very important topological invariant of arrangements of lines. This invariant was used by Chisini [1], Kulikov =-=[12]-=- and KulikovTeicher [13] in order to distinguish between connected components of the moduli space of surfaces of general type. Moreover, the Zariski-Lefschetz hyperplane section theorem (see [15]) sho... |

21 |
On the poincaré group of a rational plane curves
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(Show Context)
Citation Context ...also for computing the fundamental group of complements of hypersurfaces in CP N . A different direction for the need of fundamental groups’ computations is for getting more examples of Zariski pairs =-=[18, 19]-=-. A pair of plane curves is called a Zariski pair if they have the same combinatorics (i.e. the same singular points and the same arrangement of irreducible components), but their complements are not ... |

13 | Asymptotic enumeration of Kn-free graphs
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(Show Context)
Citation Context ...bgraph of G(L), where K4 is the full graph on 4 vertices). Note that if L is the Ceva arrangement (see Figure 1), then G(L) = K4. Some discussions on the enumeration of Kn-free graphs can be found in =-=[4, 11]-=-. Our main result is a first step in the way of proving the conjecture: Proposition 1.5. The fundamental group of following family of arrangements have a conjugation-free geometric presentation: an ar... |

13 | The fundamental group’s structure of the complement of some configurations of real line arrangements, Complex Analysis and Algebraic Geometry, edited by T
- Garber, Teicher
- 2000
(Show Context)
Citation Context ... β(L) is the first Betti number of the graph G(L) (i.e. the graph G(L) has no cycles). Then: π1(CP 2 − L) ∼ = Z r ⊕ Fm(a1)−1 ⊕ · · · ⊕ Fm(ap)−1 where r = n + p − 1 − m(a1) − · · · − m(ap). In [7] and =-=[8]-=-, similar results were achieved for the affine and projective fundamental groups in different methods. Fan [6] has conjectured that the inverse implication is also correct, i.e. if the fundamental gro... |

11 |
identità birazionale delle funzioni algebriche di due variabili dotate di una medesima curva di diramazione
- Chisini, Sulla
- 1944
(Show Context)
Citation Context ...e multiple points are of multiplicity 3. 1. Introduction The fundamental group of the complement is a very important topological invariant of arrangements of lines. This invariant was used by Chisini =-=[1]-=-, Kulikov [12] and KulikovTeicher [13] in order to distinguish between connected components of the moduli space of surfaces of general type. Moreover, the Zariski-Lefschetz hyperplane section theorem ... |

10 |
Direct product of free groups as the fundamental group of the complement of a union of lines
- Fan
- 1997
(Show Context)
Citation Context ... which there are at least two multiple points. If two such lines happen to intersect in a simple point, it is ignored (and the lines are not considered to intersect in the graph theoretic sense). Fan =-=[5, 6]-=- proved some results for the projective fundamental group: Proposition 1.1 (Fan). Let L be an arrangement of n lines and S = {a1, · · · , ap} be the set of all intersection points of L with multiplici... |

9 |
Position of singularities and fundamental group of the complement of a union of lines
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(Show Context)
Citation Context ... which there are at least two multiple points. If two such lines happen to intersect in a simple point, it is ignored (and the lines are not considered to intersect in the graph theoretic sense). Fan =-=[5, 6]-=- proved some results for the projective fundamental group: Proposition 1.1 (Fan). Let L be an arrangement of n lines and S = {a1, · · · , ap} be the set of all intersection points of L with multiplici... |

9 | Asymptotic enumeration and a 0-1 law for m-clique free graphs
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- 1985
(Show Context)
Citation Context ...bgraph of G(L), where K4 is the full graph on 4 vertices). Note that if L is the Ceva arrangement (see Figure 1), then G(L) = K4. Some discussions on the enumeration of Kn-free graphs can be found in =-=[4, 11]-=-. Our main result is a first step in the way of proving the conjecture: Proposition 1.5. The fundamental group of following family of arrangements have a conjugation-free geometric presentation: an ar... |

7 |
personal communication
- Riley
- 1995
(Show Context)
Citation Context ...l group can be read directly from the arrangement without any computation.CONJUGATION-FREE GEOMETRIC GROUPS OF ARRANGEMENTS 3 Another importance is followed by an interesting result of Eran Liberman =-=[14]-=-: Proposition 1.3. Let L be a real line arrangement with n lines, and let G = π1(C 2 − L) be the fundamental group of its affine complement. Let Γ1, . . .,Γn be the topological generators of G. Let ˜ ... |

4 | Classes of wiring diagrams and their invariants
- Garber, Teicher, et al.
(Show Context)
Citation Context ... we have a drawing of an arrangement which has a unique cycle of multiple points of length 3 as in Figure 2. We can assume it since there are several actions which do not change the fundamental group =-=[9]-=-. 4 3 6 2 n 5 n 1 Figure 2. The drawing of the arrangement with a cycle of length 3 In Block 1, one can assume that all the intersections of any horizontal line are adjacent. In Blocks 4 and 5, one ca... |

3 |
Characterization of line arrangements for which the fundamental group of the complement is a direct product, Alg
- Eliyahu, Liberman, et al.
- 2010
(Show Context)
Citation Context ... correct, i.e. if the fundamental group π1(CP 2 − L) can be written as a direct sum of free groups and infinite cyclic groups, then the graph G(L) has no cycles. Eliyahu, Liberman, Schaps and Teicher =-=[3]-=- proved this conjecture. Following these results, one can write the fundamental group by means of generators and relations, where the generators are the geometric ones (i.e. the meridians of lines), a... |

3 | On the connection between affine and projective fundamental groups of line arrangements and curves
- Garber
- 2005
(Show Context)
Citation Context ...0, where β(L) is the first Betti number of the graph G(L) (i.e. the graph G(L) has no cycles). Then: π1(CP 2 − L) ∼ = Z r ⊕ Fm(a1)−1 ⊕ · · · ⊕ Fm(ap)−1 where r = n + p − 1 − m(a1) − · · · − m(ap). In =-=[7]-=- and [8], similar results were achieved for the affine and projective fundamental groups in different methods. Fan [6] has conjectured that the inverse implication is also correct, i.e. if the fundame... |