DOCSLIB.ORG

Finite Quotients of Surface Braid Groups and Double Kodaira Fibrations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Quotients of Surface Braid Groups and Double Kodaira Fibrations Finite quotients of surface braid groups and double Kodaira fibrations Francesco Polizzi and Pietro Sabatino To Professor Ciro Ciliberto on the occasion of his 70th birthday Abstract Let Σ1 be a closed Riemann surface of genus 1. We give an account of some results obtained in the recent papers [CaPol19, Pol20, PolSab21] and concerning what we call here pure braid quotients, namely non-abelian finite groups appearing as quotients of the pure braid group on two strands P2 ¹Σ1º. We also explain how these groups can be used in order to provide new constructions of double Kodaira fibrations. 0 Introduction A Kodaira fibration is a smooth, connected holomorphic fibration 51 : ( −! 퐵1, where ( is a compact complex surface and 퐵1 is a compact complex curve, which is not isotrivial (this means that not all its fibres are biholomorphic to each others). The genus 11 := 6¹퐵1º is called the base genus of the fibration, whereas the genus 6 := 6¹퐹º, where 퐹 is any fibre, is called the fibre genus. If a surface ( is the total space of a Kodaira fibration, we will call it a Kodaira fibred surface. For every Kodaira fibration we have 11 ≥ 2 and 6 ≥ 3, see [Kas68, Theorem 1.1]. Since the fibration is smooth, the condition on the base genus implies that ( contains no rational or elliptic curves; hence it is minimal and, by the sub-additivity of the Kodaira dimension, it is of general type, hence algebraic. arXiv:2106.01743v1 [math.GT] 3 Jun 2021 Francesco Polizzi Dipartimento di Matematica e Informatica, Università della Calabria, Ponte Bucci Cubo 30B, 87036 Arcavacata di Rende (Cosenza), Italy, e-mail: [email protected] Pietro Sabatino Via Val Sillaro 5, 00141 Roma e-mail: [email protected] 1 2 Francesco Polizzi and Pietro Sabatino Examples of Kodaira fibrations were originally constructed in [Kod67, At69] in order to show that, unlike the topological Euler characteristic, the signature f of a real manifold is not multiplicative for fibre bundles. In fact, every Kodaira fibred surface ( satisfies f¹(º ¡ 0, see for example the introduction of [LLR20], whereas f¹퐵1º = f¹퐹º = 0, and so f¹(º < f¹퐵1ºf¹퐹º. A double Kodaira surface is a compact complex surface (, endowed with a double Kodaira fibration, namely a surjective, holomorphic map 5 : ( −! 퐵1 ×퐵2 yielding, by composition with the natural projections, two Kodaira fibrations 58 : ( −! 퐵8, 8 = 1, 2. The purpose of this article is to give an account of recent results, obtained in the series of papers [CaPol19, Pol20, PolSab21], concerning the construction of some double Kodaira fibrations (that we call diagonal) by means of group-theoretical methods. Let us start by introducing the needed terminology. Let 1 ≥ 2 and = ≥ 2 be two positive integers, and let P2 ¹Σ1º be the pure braid group on two strands on a closed Riemann surface of genus 1. We say that a finite group 퐺 is a pure braid quotient of type ¹1, =º if there exists a group epimorphism i : P2 ¹Σ1º −! 퐺 (1) such that i¹퐴12º has order =, where 퐴12 is the braid corresponding, via the isomor- phism P2 ¹Σ1º ' c1 ¹Σ1 × Σ1 − Δº, to the homotopy class in Σ1 × Σ1 − Δ of a loop in Σ1 × Σ1 “winding once" around the diagonal Δ. Since 퐴12 is a commutator in P2 ¹Σ1º and = ≥ 2, it follows that every pure braid quotient is a non-abelian group, see Remark2. By Grauert-Remmert’s extension theorem together with Serre’s GAGA, the exis- tence of a pure braid quotient as in (1) is equivalent to the existence of a Galois cover f : ( −! Σ1 ×Σ1, branched over Δ with branching order =. After Stein factorization, this yields in turn a diagonal double Kodaira fibration 5 : ( −! Σ11 × Σ12 . We have f = 5 , i.e. no Stein factorization is needed, if and only if 퐺 is a strong pure braid quotient, an additional condition explained in Definition3. We are now in a position to state our first results, see Theorems1,2,3: • If 1 ≥ 2 is an integer and ? ≥ 5 is a prime number, then both extra-special ?-groups of order ?41¸1 are non-strong pure braid quotients of type ¹1, ?º. • If 1 ≥ 2 is an integer and ? is a prime number dividing 1 ¸ 1, then both extra- special ?-groups of order ?21¸1 are pure braid quotients of type ¹1, ?º. • If a finite group 퐺 is a pure braid quotient, then j퐺j ≥ 32, with equality holding if and only if 퐺 is extra-special. Moreover, in the last case, we can explicitly compute the number of distinct quotients maps of type (1), up to the natural action of Aut¹퐺º. We believe that such results are significant because, although we know that P2 ¹Σ1º is residually ?-finite for all ? ≥ 2 (see [BarBel09, pp. 1481-1490]), it is usually tricky to explicitly describe its non-abelian finite quotients. The geometrical counterparts of the above group-theoretical statements allow us to construct infinite families of double Kodaira fibrations with interesting numerical Finite quotients of surface braid groups and double Kodaira fibrations 3 properties, for instance having slope greater than 2 ¸ 1/3 or signature equal to 16, see Theorems4,5,6: • Let 5 : ( ? −! Σ10 ×Σ10 be the diagonal double Kodaira fibration associated with a non-strong pure braid quotient i : P2 ¹Σ2º −! 퐺 of type ¹2,?º, where 퐺 is an extra-special ?-group 퐺 of order ?9 and 10 = ?4 ¸ 1. Then the maximum slope a¹( ?º is attained for precisely two values of ?, namely 12 a¹( º = a¹( º = 2 ¸ . 5 7 35 Furthermore, a¹( ?º ¡ 2 ¸ 1/3 for all ? ≥ 5. More precisely, if ? ≥ 7 the function a¹( ?º is strictly decreasing and 1 lim a¹( ?º = 2 ¸ . ?!¸1 3 • Let Σ1 be any closed Riemann surface of genus 1. Then there exists a double Kodaira fibration 5 : ( −! Σ1 × Σ1. Moreover, denoting by ^¹1º the number of such fibrations, we have ^¹1º ≥ !¹1 ¸ 1º, where !: N −! N stands for the arithmetic function counting the number of distinct prime factors of a positive integer. In particular, lim sup ^¹1º = ¸1. 1!¸1 • Let 퐺 be a finite group and f : ( −! Σ1 ×Σ1 be a Galois cover, with Galois group 퐺, branched over the diagonal Δ with branching order =. Then j퐺j ≥ 32, and equality holds if and only if 퐺 is extra-special. If 퐺 is extra-special of order 32 and ¹1, =º = ¹2, 2º, then f : ( −! Σ2 × Σ2 is a diagonal double Kodaira fibration such that 11 = 12 = 2, 61 = 62 = 41, f¹(º = 16. As a consequence of the last result, we obtain a sharp lower bound for the signature of a diagonal double Kodaira fibration, see Theorem7: • Let 5 : ( −! Σ11 × Σ12 be a diagonal double Kodaira fibration, associated with a pure braid quotient i : P2 ¹Σ1º −! 퐺 of type ¹1, =º. Then f¹(º ≥ 16, and equality holds precisely when ¹1, =º = ¹2, 2º and 퐺 is an extra-special group of order 32. Note that our methods show that every curve of genus 1 (and not only some special curve with extra automorphisms) is the basis of a (double) Kodaira fibration and that, in addition, the number of distinct Kodaira fibrations over a fixed base can be arbitrarily large. Furthermore, every curve of genus 2 is the base of a (double) Kodaira fibration with signature 16 and this provides, to our knowledge, the first example of positive-dimensional family of (double) Kodaira fibrations with small signature. 4 Francesco Polizzi and Pietro Sabatino The aforementioned examples with signature 16 also provide new “double so- lutions” to a problem, posed by G. Mess and included in Kirby’s problem list in low-dimensional topology, see [Kir97, Problem 2.18 A], asking what is the smallest number 1 for which there exists a real surface bundle over a real surface with base genus 1 and non-zero signature. We actually have 1 = 2, also for double Kodaira fibrations, see Theorem8: • Let ( be double Kodaira surface, associated with a pure braid quotient i : P2 ¹Σ1º −! 퐺 of type ¹2, 2º, where 퐺 is an extra-special group of order 32. Then the real manifold - underlying ( is a closed, orientable 4-manifold of signature 16 that can be realized as a real surface bundle over a real surface of genus 2, with fibre genus 41, in two different ways. In fact, it is an interesting question whether 16 and 41 are the minimum possible values for the signature and the fibre genus of a (not necessarily diagonal) double Kodaira surface 5 : ( −! Σ2 × Σ2, but we will not develop this point here. The above results paint a rather clear picture regarding pure braid quotients and the relative diagonal double Kodaira fibrations when j퐺j ≤ 32. It is natural then to investigate further this topic for j퐺j ¡ 32, and indeed this paper also contains the following new result, see Theorem9: • If 퐺 is a finite group with 32 < j퐺j < 64, then 퐺 is not a pure braid quotient. We provide only a sketch of the proof, which is based on calculations performed by means of the computer algebra system GAP4, see [GAP4]; the details will appear in a forthcoming paper. Acknowledgments. F. Polizzi was partially supported by GNSAGA-INdAM. Both authors thank A. Causin for drawing the figures. Notation and conventions. The order of a finite group 퐺 is denoted by j퐺j. If G 2 퐺, the order of G is denoted by >¹Gº. The subgroup generated by G1,...,G= 2 퐺 is denoted by hG1,...,G=i. The center of 퐺 is denoted by / ¹퐺º and the centralizer of an element G 2 퐺 by 퐶퐺 ¹Gº.
Load more

Recommended publications
  • A Classification of Clifford Algebras As Images of Group Algebras of Salingaros Vee Groups
    DEPARTMENT OF MATHEMATICS TECHNICAL REPORT A CLASSIFICATION OF CLIFFORD ALGEBRAS AS IMAGES OF GROUP ALGEBRAS OF SALINGAROS VEE GROUPS R. Ablamowicz,M.Varahagiri,A.M.Walley November 2017 No. 2017-3 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505 A Classification of Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups Rafa lAb lamowicz, Manisha Varahagiri and Anne Marie Walley Abstract. The main objective of this work is to prove that every Clifford algebra C`p;q is R-isomorphic to a quotient of a group algebra R[Gp;q] modulo an ideal J = (1 + τ) where τ is a central element of order 2. p+q+1 Here, Gp;q is a 2-group of order 2 belonging to one of Salingaros isomorphism classes N2k−1;N2k; Ω2k−1; Ω2k or Sk. Thus, Clifford al- gebras C`p;q can be classified by Salingaros classes. Since the group algebras R[Gp;q] are Z2-graded and the ideal J is homogeneous, the quotient algebras R[G]=J are Z2-graded. In some instances, the isomor- ∼ phism R[G]=J = C`p;q is also Z2-graded. By Salingaros Theorem, the groups Gp;q in the classes N2k−1 and N2k are iterative central products of the dihedral group D8 and the quaternion group Q8, and so they are extra-special. The groups in the classes Ω2k−1 and Ω2k are central products of N2k−1 and N2k with C2 × C2, respectively. The groups in the class Sk are central products of N2k or N2k with C4. Two algorithms to factor any Gp;q into an internal central product, depending on the class, are given.
    [Show full text]
  • On Sylow 2-Subgroups of Finite Simple Groups of Order up to 210
    ON SYLOW 2-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO 210 DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Sergey Malyushitsky, M.S. ***** The Ohio State University 2004 Dissertation Committee: Approved by Professor Koichiro Harada, Adviser Professor Akos Seress Adviser Professor Ronald Solomon Department of Mathematics ABSTRACT A 2-group ( a group of order a power of 2 ) is called realizable if it occurs as a Sylow 2-subgroup of a finite simple group. The purpose of this thesis is to study all realizable groups of order up to 210. From the classification of all simple groups of finite order we know all realizable groups of order up to 210 as Sylow 2-subgroups of known finite simple groups. However without the use of classification determining all realizable 2-groups is very difficult. In the first part of the thesis we present an argument that produces all realizable groups of order up to 32, by eliminating one by one all 2-groups that can not occur as a Sylow 2-subgroup of a finite simple group. When the number of 2-groups of given order becomes too large to handle it by hand we attempt to use a computer for repetitive checks on a large number of 2-groups. The second part of the thesis is devoted to describing all realizable 2-groups of order up to 210 using the classification of all finite simple groups. We determine the identification number of each group in the Small Groups Library in GAP4 and compute the power-commutator presentation of each realizable group S of type G.
    [Show full text]
  • Threads Through Group Theory
    Contemporary Mathematics Threads Through Group Theory Persi Diaconis Abstract. This paper records the path of a letter that Marty Isaacs wrote to a stranger. The tools in the letter are used to illustrate a different way of studying random walk on the Heisenberg group. The author also explains how the letter contributed to the development of super-character theory. 1. Introduction Marty Isaacs believes in answering questions. They can come from students, colleagues, or perfect strangers. As long as they seem serious, he usually gives it a try. This paper records the path of a letter that Marty wrote to a stranger (me). His letter was useful; it is reproduced in the Appendix and used extensively in Section 3. It led to more correspondence and a growing set of extensions. Here is some background. I am a mathematical statistician who, for some unknown reason, loves finite group theory. I was trying to make my own peace with a corner of p-group theory called extra-special p-groups. These are p-groups G with center Z(G) equal to commutator subgroup G0 equal to the cyclic group Cp. I found considerable literature about the groups [2, Sect. 23], [22, Chap. III, Sect. 13], [39, Chap. 4, Sect. 4] but no \stories". Where do these groups come from? Who cares about them, and how do they fit into some larger picture? My usual way to understand a group is to invent a \natural" random walk and study its rate of convergence to the uniform distribution. This often calls for detailed knowledge of the conjugacy classes, characters, and of the geometry of the Cayley graph underlying the walk.
    [Show full text]
  • The Existence of Regular Orbits for Nilpotent Groups*
    JOURNAL OF ALGEBRA 72, 54-100 (1981) The Existence of Regular Orbits for Nilpotent Groups* BEVERLY BAILEY HARGRAVES Department of Mathematical Sciences, University of Minnesota-Duluth, Duluth, Minnesota 55812 Communicated by W. Feit Received July 28. 1980 Consider the following situation. Suppose R a CR where R is an r-group and C is a cyclic group of order qe; where r and q are distinct primes. Let k be a finite field of characteristic p and suppose V is a faithful, irreducible k[CR ] module. This situation has been widely studied. When p = q it is similar to the situation occurring in the proof of Theorem B of Hall and Higman [9]. When p # q it is similar to the situation occurring in the corresponding theorem of Shult [ 131. In this paper a generalization of this situation is considered. Suppose G 4 AG where G is a solvable group, A is nilpotent and (IA /, ] G]) = 1. Again we assume V is a faithful, irreducible k[AG] module. Two related questions arise in the study of this situation. (1) When does A have a regular orbit on the elements of v#? (2) When does VI, contain the regular A module? In this paper the first question is considered under the additional assumption that char k&4 I. The following theorem (8.2) is obtained. THEOREM. Suppose G a AG where G is a solvable group, A is nilpotent and ([A 1,ICI) = 1. Let V be a faithful, irreducible k[AG] module, k afield such that char k j IA I.
    [Show full text]
  • Novel Noncommutative Cryptography Scheme Using Extra Special Group
    Hindawi Security and Communication Networks Volume 2017, Article ID 9036382, 21 pages https://doi.org/10.1155/2017/9036382 Research Article Novel Noncommutative Cryptography Scheme Using Extra Special Group Gautam Kumar and Hemraj Saini Department of Computer Science & Engineering, Jaypee University of Information Technology, Solan 173234, India Correspondence should be addressed to Gautam Kumar; [email protected] Received 19 July 2016; Revised 19 October 2016; Accepted 24 October 2016; Published 12 January 2017 Academic Editor: Pino Caballero-Gil Copyright © 2017 G. Kumar and H. Saini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Noncommutative cryptography (NCC) is truly a fascinating area with great hope of advancing performance and security for high end applications. It provides a high level of safety measures. The basis of this group is established on the hidden subgroup or subfield problem (HSP). The major focus in this manuscript is to establish the cryptographic schemes on the extra special group (ESG). ESG is showing one of the most appropriate noncommutative platforms for the solution of an open problem. The working principle is based on the random polynomials chosen by the communicating parties to secure key exchange, encryption-decryption, and authentication schemes. This group supports Heisenberg, dihedral order, and quaternion group. Further, this is enhanced from the general group elements to equivalent ring elements, known by the monomials generations for the cryptographic schemes. In this regard, special or peculiar matrices show the potential advantages.
    [Show full text]
  • The Nonexistence of a Certain Simple Group
    JOURNAL OF ALGEBRA 39, 138-149 (1976) The Nonexistence of a Certain Simple Group N. J. PATTERSON Cambridge University, England AND s. K. wONG* Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 Communicated by Walter Feit Received December 8, 1974 1. IN~~DucTI~N Let Sx denote the Suzuki’s sporadic simple group of order 44834597600 and t, an involution in the center of a Sylow 2-subgroup of Sz. Then, C, = C&to) is an extension of an extra-special group of order 2’ by the group p&(4,3). In PI, a characterization of Sx was given in terms of the group C,, . In this paper, we show in fact that there are precisely two non- isomorphic faithful extensionsof an extra-specialgroup of order 2’ byPS,(4,3). One of them, of course, is isomorphic to C, , and the other is denoted through- out this paper by C, (see Lemma 2.6). It is of interest to determine whether there is a simple group G with an involution t such that Co(t) = C, . To this end, we prove the following theorem. THEOREM A. Let G be aJinite group that possesses an involution t such that the centralizer CG(t) of t in G is isomorphic to the group C, . Then, G = O(G)&(t), where O(G) denotes the largest normal subgroup of odd order in G. An immediate consequenceof this theorem, Lemma 2.6, and [6, Main Theorem], is the following theorem. THEOREM B. Let G be a$nitegroup in which O(G) = Z(G) = 1.
    [Show full text]
  • Automorphisms of Extra Special Groups and Nonvanishing Degree 2 Cohomology
    Pacific Journal of Mathematics AUTOMORPHISMS OF EXTRA SPECIAL GROUPS AND NONVANISHING DEGREE 2 COHOMOLOGY ROBERT L. GRIESS,JR. Vol. 48, No. 2 April 1973 PACIFIC JOURNAL OF MATHEMATICS Vol. 48, No. 2, 1973 AUTOMORPHISMS OF EXTRA SPECIAL GROUPS AND NONVANISHING DEGREE 2 COHOMOLOGY ROBERT L. GRIESS, JR. If E is an extra-special 2-group, it is known that Aut (E)l Inn (E) is isomorphic to an orthogonal group. We prove that this extension is nonsplit, except in small cases. As a con- sequence, the nonvanishing of the second cohomology groups of certain classical groups (defined over F2) on their standard modules may be inferred. Also, a criterion for a subgroup of these orthogonal groups to have a nonsplit extension over the standard module is given. 1* Introduction. Let E be an extra-special 2-group of order 22n+1, n^l. That is, Ef = Z(E) and E/E' is elementary abelian. Any extra-special group may be expressed as a central product of dihedral groups D8 of order 8 and quaternion groups Q8 of order 8, with the central subgroup of order 2 in each factor amalgamated. The expression of E as such a central product is not unique in general because D8 o D8 ~ Q8 o Q8. However, the number of quaternion central factors is unique modulo 2 for any such expression (see [9] or [12]). The commutator quotient E/E' may be regarded as a vector space over the field of two elements F2 equipped with a quadratic form g, where q{xEr) = x2 e Er, for x e E (we identify E' with the additive group of F2).
    [Show full text]
  • The Whitehead Group of (Almost) Extra-Special P-Groups with P Odd Serge Bouc, Nadia Romero
    The Whitehead group of (almost) extra-special p-groups with p odd Serge Bouc, Nadia Romero To cite this version: Serge Bouc, Nadia Romero. The Whitehead group of (almost) extra-special p-groups with p odd. Journal of Pure and Applied Algebra, Elsevier, 2019. hal-01303316v2 HAL Id: hal-01303316 https://hal.archives-ouvertes.fr/hal-01303316v2 Submitted on 15 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Whitehead group of (almost) extra-special p-groups with p odd Serge Bouc1 and Nadia Romero2 Abstract: Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P , the subgroup Cl1(ZP ) of SK1(ZP ), in terms of a genetic basis of P . We also introduce a deflation map Cl1(ZP ) → Cl1 Z(P/N) , for a normal subgroup N of P , and show that it is always surjec- tive. Along the way, we give a new proof of the result describing the structure of SK1(ZP ), when P is an elementary abelian p- group.
    [Show full text]
  • A Characterization of the Suzuki Sporadic Simple Group of Order 448,345,497,600
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGRBRA 39, 277-286 (1976) A Characterization of the Suzuki Sporadic Simple Group of Order 448,345,497,600 N. J. PATTERSON Cambri&e University, England AND S. K. WONG* Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 Communicated by Walter Feit Received November 13, 1974 1. IN~~DUCTI~N In [lo], Suzuki gave a construction of a new simple group, denoted through- out in this paper by Sk. The group Sz has order 2r3 * 3l * Y * 7 . 11 . 13 and has precisely two conjugacy classes of involutions. If ~a is an involution in the center of a Sylow 2-subgroup of Sz, then C, = C&I-~) is an extension of an extra special group of order 27 by the group PSp(4,3). Our aim in this paper is to prove the converse of this fact. MAIN THEOREM. Let. G be a finite group in which O(G) = Z(G) = 1 (equivalently Z*(G) = 1). Suppose G possesses an involution 7 such that C = &(T) is isomorphicto the group C,, . Then G is isomorphicto Sz. Throughout the rest of the paper, G will always denote a group satisfying the assumption of the Main Theorem. The symbols 2,) E,, , A, , and S, will denote a cyclic group of order n, an elementary abelian group of order p”, the alternating group on n letters, and the symmetric group on n letters, respectively. Further, if A and B are groups, A’ B will denote an extension of a group A by a group B.
    [Show full text]
  • Mathieu Moonshine and Orbifold K3s
    Mathieu Moonshine and Orbifold K3s Matthias R. Gaberdiel and Roberto Volpato Abstract The current status of ‘Mathieu Moonshine’, the idea that the Mathieu group M24 organises the elliptic genus of K3, is reviewed. While there is a consistent decomposition of all Fourier coefficients of the elliptic genus in terms of Mathieu M24 representations, a conceptual understanding of this phenomenon in terms of K3 sigma-models is still missing. In particular, it follows from the recent classification of the automorphism groups of arbitrary K3 sigma-models that (1) there is no single K3 sigma-model that has M24 as an automorphism group; and (2) there exist ‘exceptional’ K3 sigma-models whose automorphism group is not even a subgroup of M24. Here we show that all cyclic torus orbifolds are exceptional in this sense, and that almost all of the exceptional cases are realised as cyclic torus orbifolds. We also provide an explicit construction of a Z5 torus orbifold that realises one exceptional class of K3 sigma-models. 1 Introduction In 2010, Eguchi et al. observed that the elliptic genus of K3 shows signs of an underlying Mathieu M24 group action [1]. In particular, they noted (see Sect. 2 below for more details) that the Fourier coefficients of the elliptic genus can be M.R. Gaberdiel () Institut für Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland e-mail: [email protected] R. Volpato Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14476 Golm, Germany e-mail: [email protected] W. Kohnen and R. Weissauer (eds.), Conformal Field Theory, Automorphic Forms 109 and Related Topics, Contributions in Mathematical and Computational Sciences 8, DOI 10.1007/978-3-662-43831-2__5, © Springer-Verlag Berlin Heidelberg 2014 110 M.R.
    [Show full text]
  • Arxiv:2103.07035V1 [Math.QA] 12 Mar 2021 H Riodvoa Orbifold the 0-15M01-0 M3o Taiwan
    FOURVOLUTIONS AND AUTOMORPHISM GROUPS OF ORBIFOLD LATTICE VERTEX OPERATOR ALGEBRAS HSIAN-YANG CHEN∗ AND CHING HUNG LAM Abstract. Let L be an even positive definite lattice with no roots, i.e., L(2) = x L { ∈ | (x x)=2 = . Let g O(L) be an isometry of order 4 such that g2 = 1 on L. In this | } ∅ ∈ − article, we determine the full automorphism group of the orbifold vertex operator algebra V gˆ. As our main result, we show that Aut (V gˆ) is isomorphic to N ( gˆ )/ gˆ unless L L Aut (VL) h i h i √ L ∼= 2E8 or BW16. 1. Introduction Let V be a vertex operator algebra (abbreviated as VOA) and let g Aut V be a finite ∈ automorphism of V . The fixed-point subalgebra V g = v V gv = v { ∈ | } is often called an orbifold subVOA. Our main propose is to study the full automorphism group of the orbifold VOA V g. It is clear that the normalizer N ( g ) of g stabilizes Aut (V ) h i the orbifold VOA V g and it induces a group homomorphism f : N ( g )/ g Aut (V ) h i h i → Aut (V g). For generic cases, Aut (V g) is often isomorphic to N ( g )/ g but Aut (V g) Aut (V ) h i h i may be strictly bigger than N ( g )/ g . We call an automorphism s Aut (V g) an Aut (V ) h i h i ∈ arXiv:2103.07035v1 [math.QA] 12 Mar 2021 extra automorphism if s is not in the image of f, or equivalently, s is not a restriction from an automorphism in Aut(V ).
    [Show full text]
  • A Study of Extra Special P-Group 1 2 MURTADHA ALI SHABEEB , DR
    ISSN 2319-8885 Vol.02,Issue.19, December-2013, Pages:2223-2234 www.semargroups.org, www.ijsetr.com A Study of Extra Special P-Group 1 2 MURTADHA ALI SHABEEB , DR. SWAPNIL SRIVASTAVA 1MSc, Dept of Mathematics, SHITAS, Allahabad, UP-INDIA, Email: [email protected]. 2Asst Prof, Dept of Mathematics, SHIATS, Allahabad, UP-INDIA. Abstract: In this dissertation we have discussed extra special -group. A finite non- abelian -group is called extra special - group if its center is exactly equal to its commutator subgroup. Here we have discussed extra special -groups and we have find that every non-abelian group of order is extra special -group. In particular if then we have two extra special -groups one is dihedral group and another is Hamiltonian group . Here we have also discussed that if is non-abelian group of order , then has order . We have thoroughly discussed the following theorem: Let be a finite extra special group. Then it is central product of non-abelian groups of order . In particular is of order for some . To prove above theorem we have gone through solvability and nilpotency in groups, Frattini subgroups, and different type of bilinear forms. Keywords: P-Group, Hamiltonian Group. I. INTRODUCTION only if the structure ( namely the Galois group) possess a Algebra is one of the broad parts of Mathematics, property ( called the solvability ). In the second half of together with number theory, geometry and analysis. For the nineteenth century the notion of the congruences of historical reasons, the word “algebra” has several related Geometric objects was generalized. The development meaning in mathematics, as a single word or with qualifies.
    [Show full text]