The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

On Projective Plane of Order 13 with a Frobenius Group of Order 39 as a Collineation Group

One of the most outstanding problems in combinatorial mathematics and geometry is the problem of existence of finite projective planes whose order is not a prime power.

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

We classify the ( finite and infinite) virtually cyclic subgroups of the pure braid groups P(n)(RP(2)) of the projective plane. The maximal finite subgroups of P(n)(RP(2)) are isomorphic to the quaternion group of order 8 if n = 3, and to Z(4) if n >= 4. Further, for all n >= 3, the following groups are, up to isomorphism, the infinite virtually cyclic subgroups of P(n)(RP(2)): Z, Z(2) x Z and the amalgamated product Z(4)*(Z2)Z(4).

THE FUNDAMENTAL GROUP OF THE SPACE OF MAPS FROM A SURFACE INTO THE PROJECTIVE PLANE

In this work we compute the fundamental group of each connected component of the function space of maps from it closed surface into the projective space

A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings.

On kuhnels 9-vertex complex projective plane

We present an elementary combinatorial proof of the existence and uniqueness of the 9-vertex triangulation of C P2. The original proof of existence, due to Kuhnel, as well as the original proof of uniqueness, due to Kuhnel and Lassmann, were based on extensive computer search. Recently Arnoux and Marin have used cohomology theory to present a computer-free proof. Our proof has the advantage of displaying a canonical copy of the affine plane over the three-element field inside this complex in terms of which the entire complex has a very neat and short description. This explicates the full automorphism group of the Kuhnel complex as a subgroup of the automorphism group of this affine plane. Our method also brings out the rich combinatorial structure inside this complex.

The projective plane, J-holomorphic curves and Desargues' theorem

By a theorem of Gromov, for an almost complex structure J on CP2 tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.

On some Finite-Dimensional Representations of Artin Braid Group

Валентин В. Илиев - Авторът изучава някои хомоморфни образи G на групата на Артин на плитките върху n нишки в крайни симетрични групи. Получените пермутационни групи G са разширения на симетричната група върху n букви чрез подходяща абелева група. Разширенията G зависят от един целочислен параметър q ≥ 1 и се разцепват тогава и само тогава, когато 4 не дели q. В случая на нечетно q са намерени всички крайномерни неприводими представяния на G, а те от своя страна генерират безкрайна редица от неприводими представяния на групата на плитките.

Algorithms for Finding Unitals and Maximal Arcs in Projective Planes of Order 16

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakalo ff , Sofia, July, 2006.

From the Icosahedron to Natural Triangulations of CP(2) and S(2) x S(2)

We present two constructions in this paper: (a) a 10-vertex triangulation CP(10)(2) of the complex projective plane CP(2) as a subcomplex of the join of the standard sphere (S(4)(2)) and the standard real projective plane (RP(6)(2), the decahedron), its automorphism group is A(4); (b) a 12-vertex triangulation (S(2) x S(2))(12) of S(2) x S(2) with automorphism group 2S(5), the Schur double cover of the symmetric group S(5). It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of S(2) x S(2). Both constructions have surprising and intimate relationships with the icosahedron. It is well known that CP(2) has S(2) x S(2) as a two-fold branched cover; we construct the triangulation CP(10)(2) of CP(2) by presenting a simplicial realization of this covering map S(2) x S(2) -> CP(2). The domain of this simplicial map is a simplicial subdivision of the standard cell structure of S(2) x S(2), different from the triangulation alluded to in (b). This gives a new proof that Kuhnel's CP(9)(2) triangulates CP(2). It is also shown that CP(10)(2) and (S(2) x S(2))(12) induce the standard piecewise linear structure on CP(2) and S(2) x S(2) respectively.

Minimal triangulation, complementarity and projective planes

Brehm and Kuhnel proved that if M-d is a combinatorial d-manifold with 3d/2 + 3 vertices and \ M-d \ is not homeomorphic to Sd then the combinatorial Morse number of M-d is three and hence d is an element of {0, 2, 4, 8, 16} and \ M-d \ is a manifold like a projective plane in the sense of Eells and Kuiper. We discuss the existence and uniqueness of such combinatorial manifolds. We also present the following result: ''Let M-n(d) be a combinatorial d-manifold with n vertices. M-n(d) satisfies complementarity if and only if d is an element of {0, 2, 4, 8, 16} with n = 3d/2 + 3 and \ M-n(d) \ is a manifold like a projective plane''.

Action de groupe, formes normales et systèmes quadratiques à foyer faible d'ordre trois

Dans ce mémoire, on s'intéresse à l'action du groupe des transformations affines et des homothéties sur l'axe du temps des systèmes différentiels quadratiques à foyer faible d'ordre trois, dans le plan. Ces systèmes sont importants dans le cadre du seizième problème d'Hilbert. Le diagramme de bifurcation a été produit à l'aide de la forme normale de Li dans des travaux de Andronova [2] et Artès et Llibre [4], sans utiliser le plan projectif comme espace des paramètres ni de méthodes globales. Dans [7], Llibre et Schlomiuk ont utilisé le plan projectif comme espace des paramètres et des notions à caractère géométrique global (invariants affines et topologiques). Ce diagramme contient 18 portraits de phase et certains de ces portraits sont répétés dans des parties distinctes du diagramme. Ceci nous mène à poser la question suivante : existe-t-il des systèmes distincts, correspondant à des valeurs distinctes de paramètres, se trouvant sur la même orbite par rapport à l'action du groupe? Dans ce mémoire, on prouve un résultat original : l'action du groupe n'est pas triviale sur la forme de Li (théorème 3.1), ni sur la forme normale de Bautin (théorème 4.1). En utilisant le deuxième résultat, on construit l'espace topologique quotient des systèmes quadratiques à foyer faible d'ordre trois par rapport à l'action de ce groupe.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell-Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n >= 1, the class of m-string braid groups B(m)(S(2)\{x(1), ... , x(n)}) of the n-punctured sphere includes the usual Artin braid groups B(m) (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (C) over tilde (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B(m) is completely determined for all m is an element of N (respectively, for all m not equal 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n >= 2, we prove that the lower central series (respectively, derived series) of B(m)(S(2)\{x(1), ... , x(n)}) is constant from the commutator subgroup onwards for all m >= 3 (respectively, m >= 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B(2)(S(2)\{x(1), x(2)}) admits various interpretations, as the Baumslag-Solitar group BS(2, 2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B(2)(S(2)\{x(1), x(2)}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product Z(2) * Z. Further, its lower central series quotients Gamma(i)/Gamma(i+1) are direct sums of copies of Z(2), the number of summands being determined explicitly. In the case m >= 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Gamma(i)/Gamma(i+1) are 2-elementary finitely-generated groups.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n >= 3. We classify the finite groups which are realised as subgroups of the sphere braid group B(n)(S(2)). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B(n)(S(2)): Z(2(n-1)); the dicyclic groups of order 4n and 4(n - 2); the binary tetrahedral group T*; the binary octahedral group O*; and the binary icosahedral group I(*). We give geometric as well as some explicit algebraic constructions of these groups in B(n)(S(2)) and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi`s classification of the torsion elements of B(n)(S(2)) and explain how the finite subgroups of B(n)(S(2)) are related to this classification, as well as to the lower central and derived series of B(n)(S(2)).