Automorphisms of Riemann Surfaces, Subgroups of Mapping Class Groups and Related Topics

• Record Nr.: UNINA9910959086703321
• Autore: Wootton Aaron
• Pubbl/distr/stampa: Providence : , : American Mathematical Society, , 2022; ©2022
• ISBN: 9781470467760; 1470467763
• Edizione: [1st ed.]
• Descrizione fisica: 1 online resource (366 pages)
• Collana: Contemporary Mathematics ; ; v.776
• Classificazione: 30Fxx14Hxx20H1020B2511G3257K20
• Altri autori (Persone): BroughtonS. Allen; PaulhusJennifer
• Disciplina: 515/.93
• Soggetti: Riemann surfaces; Automorphisms; Group theory; Functions of a complex variable -- Riemann surfaces; Algebraic geometry -- Curves in algebraic geometry; Group theory and generalizations -- Other groups of matrices -- Fuchsian groups and their generalizations (group-theoretic aspects); Group theory and generalizations -- Permutation groups -- Finite automorphism groups of algebraic, geometric, or combinatorial structures; Number theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Arithmetic aspects of dessins d'enfants, Belyĭ theory; Manifolds and cell complexes -- Low-dimensional topology in specific dimensions -- 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Cover -- Title page -- Contents -- Preface -- The engaging symmetry of Riemann surfaces: A historical perspective -- 1. Introduction -- 2. Compact Riemann surfaces and their automorphisms -- 3. Hurwitz surfaces and groups and other interesting families -- 4. Maps and hypermaps -- 5. Dessins d'enfants and quasiplatonic surfaces -- 6. Connections with Galois theory -- References -- Future directions in ; automorphisms of surfaces, graphs, and other related topics -- 1. Introduction -- 2. Preliminaries -- 2.1. Conformal group actions on surfaces and their construction -- 2.1.1. Monodromy epimorphisms -- 2.1.2. Surface kernel epimorphisms -- 2.2. Equivalence of actions -- 3. Automorphism groups of Riemann surfaces -- 3.1. Classification results -- 3.2. Defining equations for surfaces and automorphisms -- 3.3. The genus spectrum of a group -- 3.4. Relationship with subgroups of mapping class groups -- 3.5. Full automorphism groups and maximal group orders -- 3.6. Signature realization -- 4. Families of Riemann surfaces and their moduli -- 4.1. Hurwitz spaces -- 4.2. Moduli and Teichmüller spaces -- 5. Curves -- 5.1. Extending results from hyperelliptic and superelliptic curves -- 5.2. Jacobian varieties -- 6. Graphs, dessins d'enfant and quasiplatonic surfaces -- 6.1. Dessins d'enfants -- 6.2. Quasiplatonic surfaces -- 6.3. Building surfaces and actions from a tiling -- 6.4. Further directions -- 7. Symmetries of surfaces -- 7.1. Symmetries of quasiplatonic surfaces -- 7.2. Existence of symmetries -- 7.3. Symmetric -gonal actions -- 8. Algorithms, computations, and explicit methods -- 8.1. Classifications -- 8.2. Equivalence relations -- 8.3. Problems on enumerating actions -- 9. Acknowledgments -- References -- Extending Harvey's surface kernel maps -- 1. Introduction -- 2. History and Methods -- 3. Notation -- 4. Harvey's results and Extensions.; 5. Background: Summary of the Reidemeister-Schreier theory -- 6. Application of the Reidemeister-Schreier Theorem -- 7. The symmetric group ₃ and its multiplication table -- 8. Detailed Calculations for ₃ -- 9. Questions -- 10. Acknowledgments -- References -- A short proof of Greenberg's Theorem -- 1. Introduction -- 2. The proof -- 3. Remarks -- References -- Equivalence of finite group actions on Riemann surfaces and algebraic curves -- 1. Motivation and overview -- 2. Preliminaries -- 3. Rotation data of a -action -- 4. Equivalence of actions -- 5. Comparison of equivalence relations -- 6. Analysis of Conflation -- References -- Planar representations of group actions on surfaces -- 1. Introduction -- 2. Preliminaries -- 3. Skeletal uniqueness property -- 4. Skeletal signature spaces of SUP Groups: the strategy -- 5. The -SUP group case -- 6. The -SUP group case -- 7. The ²-SUP group case for odd -- 8. The 4-SUP group case -- Acknowledgments -- References -- Fiber product of Riemann surfaces -- 1. Introduction -- 2. The fiber product of Riemann surfaces -- 3. The strong field of moduli of the fiber product -- 4. Isogenous decomposition of the Jacobian variety of fiber products -- 5. Examples -- References -- One dimensional equisymmetric strata in moduli space -- 1. Introduction -- 2. Preliminaries -- 3. Covering -gonal strata by Hurwitz spaces -- 4. Case: Orbit genus 0 and 4 branch points -- References -- Arithmetic of dihedral origami -- 1. Introduction -- 2. Elliptic curve and origami set-up -- 3. Construction -- 4. Division polynomials -- 5. Galois representations -- 6. Diagram -- References -- Reduction of superelliptic Riemann surfaces -- 1. Introduction -- 2. Preliminaries -- 3. Reduction of the moduli point -- 4. Reduction of coefficients of binary forms -- 5. Concluding remarks -- References.; Dessins d'enfants with a given bipartite graph -- 1. Introduction -- 2. Preliminaries -- 3. Proof of Theorem 1 -- 4. Some classical bipartite graphs -- Acknowledgment -- References -- On infinite octavalent polyhedral surfaces -- 1. Introduction -- 2. Background -- 3. 8(1,1,6) as Schwarz CLP minimal surface -- 4. A triangulated Swiss cross -- 5. Figure credits -- References -- Universal -gonal tessellations and their Petrie paths -- 1. Maps on surfaces -- 2. Algebraic maps -- 3. Universal maps and map subgroups -- 4. Hecke groups -- 5. Infinite periods -- 6. The universal -gonal map -- 7. Petrie paths -- 8. The ; special cases =4,6 -- 9. Even and odd vertices of Petrie paths -- 10. The principal Petrie paths of ℳ̂₅ -- References -- On the Riemann-Hurwitz formula for regular graph coverings -- 1. Introduction -- 2. Preliminary results and definitions -- 3. Groups acting on a graph without invertible edges -- 4. Groups acting on a graph with invertible edges -- Acknowledgments -- References -- Cyclic and dihedral actions on Klein surfaces with 2 boundary components -- 1. Introduction and preliminaries -- 2. The case odd -- 3. The case even: preliminary conditions -- 4. Cyclic groups -- 5. Dihedral groups -- 6. Concluding remarks -- References -- Finitely generated non-cocompact NEC groups -- 1. Introduction -- 2. Preliminaries -- 3. Surface symbols -- 4. Canonical presentation -- References -- Back Cover.
• Sommario/riassunto: Automorphism groups of Riemann surfaces have been widely studied for almost 150 years. This area has persisted in part because it has close ties to many other topics of interest such as number theory, graph theory, mapping class groups, and geometric and computational group theory. In recent years there has been a major revival in this area due in part to great advances in computer algebra systems and progress in finite group theory.This volume provides a concise but thorough introduction for newcomers to the area while at the same time highlighting new developments for established researchers. The volume starts with two expository articles. The first of these articles gives a historical perspective of the field with an emphasis on highly symmetric surfaces, such as Hurwitz surfaces. The second expository article focuses on the future of the field, outlining some of the more popular topics in recent years and providing 78 open research problems across all topics. The remaining articles showcase new developments in the area and have specifically been chosen to cover a variety of topics to illustrate the range of diversity within the field.