Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
Beauville p-Groups: A Survey
Published online by Cambridge University Press: 15 April 2019
Book contents
- Frontmatter
- Contents
- Introduction
- Finite Simple Groups and Fusion Systems
- Finite and Infinite Quotients of Discrete and Indiscrete Groups
- Local-Global Conjectures and Blocks of Simple Groups
- A Survey on Some Methods of Generating Finite Simple Groups
- One-Relator Groups: An Overview
- New Progress in Products of Conjugacy Classes in Finite Groups
- Aspherical Relative Presentations all Over Again
- Simple Groups, Generation and Probabilistic Methods
- Irreducible Subgroups of Simple Algebraic Groups – A Survey
- Practical Computation with Linear Groups Over Infinite Domains
- Beauville p-Groups: A Survey
- Structural Criteria in Factorised Groups Via Conjugacy Class Sizes
- Growth in Linear Algebraic Groups and Permutation Groups: Towards a Unified Perspective
- L2-Betti Numbers and their Analogues in Positive Characteristic
- On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
- Vertex Stabilizers of Graphs with Primitive Automorphism Groups and a Strong Version of the Sims Conjecture
- On the Character Degrees of a Sylow p-Subgroup of a Finite Chevalley Group G(pf) Over a Bad Prime
- Patterns on Symmetric Riemann Surfaces
- Subgroups of Twisted Wreath Products
- Some Remarks on Self-Dual Codes Invariant Under Almost Simple Permutation Groups
- Test Elements: From Pro-p to Discrete Groups
- References
Summary
Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the p-groups that may be used in this way. En route we discuss several open problems, questions and conjectures.
- Type
- Chapter
- Information
- Groups St Andrews 2017 in Birmingham , pp. 271 - 288Publisher: Cambridge University PressPrint publication year: 2019
References
Barker, N. W., Boston, N. and Fairbairn, B. T., A note on Beauville p-groups, Exp. Math. 21 (2012), no. 3, 298–306.CrossRefGoogle Scholar
Barker, N. W., Boston, N., Peyerimhoff, N. and Vdovina, A., New examples of Beauville surfaces, Monatsh. Math. 166 (2012), no. 3–4, 319–327.CrossRefGoogle Scholar
Barker, N. W., Boston, N., Peyerimhoff, N. and Vdovina, A., An infinite family of 2-groups with mixed Beauville structures, Int. Math. Res. Not. IMRN 2015, no. 11, 3598–3618.CrossRefGoogle Scholar
Barker, N. W., Boston, N., Peyerimhoff, N. and Vdovina, A., Regular algebraic surfaces, ramification structures and projective planes, in Beauville Surfaces and Groups (eds. Bause, I., Garion, S. and Vdovina, A.), 15–33, Springer Proc. Math. Stat., 123, Springer, 2015.Google Scholar
Bauer, I., Product-Quotient Suraces: Result and Problems, preprint 2012. arxiv:1204.3409Google Scholar
Bauer, I., Catanese, F. and Frapporti, D., The fundamental group and torsion group of Beauville surfaces, in Beauville Surfaces and Groups (eds. Bauer, I. Garion, S. and Vdovina, A.), 1–14 (2015).CrossRefGoogle Scholar
Bauer, I., Catanese, F. and Grunewald, F., Beauville surfaces without real structures, in Geometric methods in algebra and number theory, 1–42, Progr. Math., 235, Birkhäuser Boston, Boston, MA, 2005.Google Scholar
Bauer, I., Catanese, F. and Grunewald, F., Chebycheff and Belyi polynomials, dessins d’enfants, Beauville surfaces and group theory, Mediterr. J. Math. 3 (2006), 121–146.CrossRefGoogle Scholar
Bauer, I., Catanese, F. and Grunewald, F., The classification of surfaces with pg = q = 0 isogenous to a product of curves, Pre Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov. Part 1, 547–586.Google Scholar
Bauer, I. C., Catanese, F. and Pignatelli, R., Complex surfaces of general type: some recent progress, in Global aspects of complex geometry, 1–58, Springer, Berlin, 2006.Google Scholar
Bauer, I., Catanese, F. and Pignatelli, R., Surfaces of general type with geometric genus zero: a survey, in Complex and differential geometry, 1–48, Springer Proc. Math., 8, Springer-Verlag, Heidelberg, 2011.Google Scholar
Beauville, A., Complex Algebraic Surfaces, London Math. Soc. Student Texts 34, Cambridge University Press, Cambridge, 1996.CrossRefGoogle Scholar
BelyĬ, G. V., On Galois extensions of a maximal cyclotomic field, Math. USSR Izv. 14 (1980), 247–256.CrossRefGoogle Scholar
Besche, H. U., Eick, B. and O’Brien, E. A., The groups of order at most 2 000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1–4.Google Scholar
Besche, H. U., Eick, B. and O’Brien, E. A., A millennium project: constructing small groups, Internat. J. Algebra Comput. 12 (2002), no. 5, 623–644.CrossRefGoogle Scholar
Boston, N., A survey of Beauville p-groups, in Beauville Surfaces and Groups (eds. Bauer, I., Garion, S. and Vdovina, A.), 35–40, Springer Proc. Math. Stat., 123, Springer, Cham, 2015.Google Scholar
Catanese, F., Fibered surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), no. 1, 1–44.CrossRefGoogle Scholar
Catanese, F., Moduli spaces of surfaces and real structures, Ann. of Math. (2) 158 (2003), no. 2, 577–592.CrossRefGoogle Scholar
Fairbairn, B. T., Strongly real Beauville groups, in Beauville Surfaces and Group (eds. Bauer, I., Garion, S. and Vdovina, A.), 41–61, Springer Proc. Math. Stat., 123, Springer, Cham, 2015.Google Scholar
Fairbairn, B. T., More on strongly real Beauville groups, in Symmetries in Graphs, Maps and Polytopes, (eds. Širáň, J. and Jajcay, R.), 129–146, Springer Proc. Math. Stat., 159, Springer, 2016.Google Scholar
Fairbairn, B. T., Recent work on Beauville surfaces, structures and groups, in Groups St Andrews 2013 (eds. Campbell, C. M., Quick, M. R., Robertson, E. F. and Roney-Dougal, C. M.), 225–241, London Math. Soc. Lecture Note Ser. 422, Cambridge University Press, 2015.Google Scholar
Fairbairn, B. T., A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p, Bull. Lond. Math. Soc. 49 (2017), no. 5, 749–654.CrossRefGoogle Scholar
Fairbairn, B. T., A Corrigendum to “Fairbairn, A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p”, in preparation.Google Scholar
Fairbairn, B. T. and Pierro, E., New examples of mixed Beauville groups, J. Group Theory 18 (2015), no. 5, 761-792.CrossRefGoogle Scholar
Fernández-Alcober, G. A., Gavioli, N., Gül, Ş. and Scoppola, C. M., Beauville p-groups of wild type and groups of maximal class, preprint 2017. arXiv1701.07361.Google Scholar
Fernández-Alcober, G. A. and Gül, Ş., Beauville structures in finite p-groups, preprint 2016. arXiv:1507.02942.CrossRefGoogle Scholar
Fuertes, Y. and González-Diez, G., On Beauville structures on the groups Sn and An, Math. Z. 264 (2010), no. 4, 959–968.CrossRefGoogle Scholar
Fuertes, Y., González-Diez, G. and Jaikin-Zapirain, A., On Beauville surfaces, Groups Geom. Dyn. 5 (2011), no. 1, 107–119.CrossRefGoogle Scholar
Fuertes, Y. and Jones, G., Beauville surfaces and finite groups, J. Algebra 340 (2011), 13–27.CrossRefGoogle Scholar
Galkin, S. and Shinder, E., Exceptional collections of line bundles on the Beauville surface, Adv. Math. 244 (2013), 1033–1050.CrossRefGoogle Scholar
Garion, S. and Penegini, M., New Beauville surfaces, moduli spaces and finite groups, Comm. Algebra 42 (2014), no. 5, 2126–2155.CrossRefGoogle Scholar
Gavioli, N., Gül, Ş. and Scoppola, C., Metabelian thin Beauville p-groups, preprint 2017. arXiv:1701.06906CrossRefGoogle Scholar
Girondo, E. and González-Diez, G., Introduction to Compact Riemann Surfaces and Dessins d’Enfants, London Math. Soc. Student Texts 79, Cambridge University Press, Cambridge 2011.CrossRefGoogle Scholar
González-Diez, G. and Jaikin-Zapirain, A., The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, Proc. London Math. Soc. 111 (2015), no. 4, 775–796.CrossRefGoogle Scholar
González-Diez, G., Jones, G. A. and Torres-Teigell, D., Beauville surfaces with abelian Beauville group, Math. Scand. 114 (2014), no. 2, 191-204.CrossRefGoogle Scholar
González-Diez, G. and Torres-Teigell, D., An introduction to Beauville surfaces via uniformization, in Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces, 123–151, Contemp. Math., 575, Amer. Math. Soc., Providence, RI, 2012.Google Scholar
González-Diez, G. and Torres-Teigell, D., Non-homeomorphic Galois conjugate Beauville structures on PSL(2, p), Adv. Math. 229 (2012), 3096–3122.CrossRefGoogle Scholar
González-Diez, G., Jones, G. A. and Torres-Teigell, D., Arbitrarily large Galois orbits of non-homeomorphic surfaces, arXiv:1110.4930.Google Scholar
Grothendieck, A., Esquisse d’un Programme, in Geometric Galois Actions 1. in Around Grothendieck’s Esquisse d’un Programme (eds. Lochak, P. and Schneps, L.), 5–84, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, 1997.Google Scholar
Gül, Ş., An infinite family of strongly real Beauville p-groups, preprint 2016. arXiv:1610.06080.CrossRefGoogle Scholar
Helleloid, G. T. and Martin, U., The automorphism group of a finite p-group is almost always a p-group, J. Algebra 312 (2007), 284–329.CrossRefGoogle Scholar
Jones, G. A., Automorphism groups of Beauville surfaces, J. Group Theory. 16 (2013), no. 3, 353–381.CrossRefGoogle Scholar
Jones, G. A., Beauville surfaces and groups: a survey, in Rigidity and Symmetry (eds. Connelly, R., Weiss, A. and Whitely, W.), 205–225, Fields Inst. Commun., 70, Springer, New York, 2014.Google Scholar
Jones, G. A., Characteristically simple Beauville groups I: Cartesian powers of alternating groups, in Geometry, groups and dynamics, 289–306, Contemp. Math., 639, Amer. Math. Soc., Providence, RI, 2015.Google Scholar
Jones, G. A., Characteristically Simple Beauville Groups, II: Low Rank and Sporadic Groups, in Beauville Surfaces and Groups (eds. Bauer, I., Garion, S. and Vdovina, A.), 97–120, Springer Proc. Math. Stat., 123, Springer 2015.Google Scholar
Jones, G. A. and Singerman, D., Belyi functions, hypermaps and Galois groups, Bull. Lond. Math. Soc. 28 (1996), 561–590.CrossRefGoogle Scholar
Jones, G. A. and Wolfart, J. Dessins d’Enfants on Riemann Surfaces, Springer Monographs in Mathematics (2016).CrossRefGoogle Scholar
O’Brien, E. A., The p-group generation algorithm, J. Symbolic Comput. 9 (1990), no. 5–6, 677–698.Google Scholar
Peyerimhoff, N. and Vdovina, A., Cayley graph expanders and groups of finite width, J. Pure Appl. Algebra 215 (2011), no. 11, 2780–2788.CrossRefGoogle Scholar
Pierro, E., Some calculations on the action of groups on surfaces, PhD thesis, Birkbeck, University of London (2015).Google Scholar
Širáň, J., How symmetric can maps on surfaces be?, in Surveys in Combinatorics 2013 (eds. Blackburn, Simon R., Gerke, Stefanie and Wildon, Mark), 161–238, London Math. Soc. Lecture Note Ser., 409, Cambridge Univ. Press, Cambridge, 2013.Google Scholar
Stix, J. and Vdovina, A., Series of p-groups with Beauville structure, Monatsh. Math. 181 (2016), no. 1, 177–186.CrossRefGoogle Scholar
Torres-Teigell, D., Triangle groups, dessins d’enfants and Beauville surfaces, PhD thesis, Universidad Autonoma de Madrid, 2012.Google Scholar
Wolfart, J., ABC for polynomials, dessins d’enfants and uniformization — a survey, in Elementare und analytische Zahlentheorie, 313–345, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006.Google Scholar
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