LEADER 09628nam 22007213 450 001 9910959086703321 005 20231110212143.0 010 $a9781470467760 010 $a1470467763 035 $a(CKB)5680000000017062 035 $a(MiAaPQ)EBC6913862 035 $a(Au-PeEL)EBL6913862 035 $a(RPAM)22253566 035 $a(OCoLC)1303085836 035 $a(EXLCZ)995680000000017062 100 $a20220310d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAutomorphisms of Riemann Surfaces, Subgroups of Mapping Class Groups and Related Topics 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (366 pages) 225 1 $aContemporary Mathematics ;$vv.776 320 $aIncludes bibliographical references. 327 $aCover -- 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. 327 $a5. Background: Summary of the Reidemeister-Schreier theory -- 6. Application of the Reidemeister-Schreier Theorem -- 7. The symmetric group ? and its multiplication table -- 8. Detailed Calculations for ? -- 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. 327 $aDessins 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 ??? -- 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. 330 $aAutomorphism groups of Riemann surfaces have been widely studied for almost 150 years. This area has persisted in part because it has close ties to many other topics of interest such as number theory, graph theory, mapping class groups, and geometric and computational group theory. In recent years there has been a major revival in this area due in part to great advances in computer algebra systems and progress in finite group theory.This volume provides a concise but thorough introduction for newcomers to the area while at the same time highlighting new developments for established researchers. The volume starts with two expository articles. The first of these articles gives a historical perspective of the field with an emphasis on highly symmetric surfaces, such as Hurwitz surfaces. The second expository article focuses on the future of the field, outlining some of the more popular topics in recent years and providing 78 open research problems across all topics. The remaining articles showcase new developments in the area and have specifically been chosen to cover a variety of topics to illustrate the range of diversity within the field. 410 0$aContemporary Mathematics 606 $aRiemann surfaces 606 $aAutomorphisms 606 $aGroup theory 606 $aFunctions of a complex variable -- Riemann surfaces$2msc 606 $aAlgebraic geometry -- Curves in algebraic geometry$2msc 606 $aGroup theory and generalizations -- Other groups of matrices -- Fuchsian groups and their generalizations (group-theoretic aspects)$2msc 606 $aGroup theory and generalizations -- Permutation groups -- Finite automorphism groups of algebraic, geometric, or combinatorial structures$2msc 606 $aNumber theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Arithmetic aspects of dessins d'enfants, Belyi? theory$2msc 606 $aManifolds and cell complexes -- Low-dimensional topology in specific dimensions -- 2-dimensional topology (including mapping class groups of surfaces, Teichmu?ller theory, curve complexes, etc.)$2msc 615 0$aRiemann surfaces. 615 0$aAutomorphisms. 615 0$aGroup theory. 615 7$aFunctions of a complex variable -- Riemann surfaces. 615 7$aAlgebraic geometry -- Curves in algebraic geometry. 615 7$aGroup theory and generalizations -- Other groups of matrices -- Fuchsian groups and their generalizations (group-theoretic aspects). 615 7$aGroup theory and generalizations -- Permutation groups -- Finite automorphism groups of algebraic, geometric, or combinatorial structures. 615 7$aNumber theory -- Arithmetic algebraic geometry (Diophantine geometry) -- Arithmetic aspects of dessins d'enfants, Belyi? theory. 615 7$aManifolds and cell complexes -- Low-dimensional topology in specific dimensions -- 2-dimensional topology (including mapping class groups of surfaces, Teichmu?ller theory, curve complexes, etc.). 676 $a515/.93 686 $a30Fxx$a14Hxx$a20H10$a20B25$a11G32$a57K20$2msc 700 $aWootton$b Aaron$01799828 701 $aBroughton$b S. Allen$01799829 701 $aPaulhus$b Jennifer$01799830 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910959086703321 996 $aAutomorphisms of Riemann Surfaces, Subgroups of Mapping Class Groups and Related Topics$94344240 997 $aUNINA