Contents. Preface... Part I Foundations for the theory of the discontinuous groups of linear substitutions of one variable

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1 Preface... XXV 0 Introduction. Developments concerning projective determinations of measure The projective determinations of measure in the plane and their divisionintokinds The motions belonging to a determination of measure and symmetric transformations of the plane into itself. The variable ζ in the parabolic case Setting up all collineations of the conic section z 1 z 3 z2 2 = 0 into itself. Behavior of the associated ζ The group of the motion and symmetric transformations for the hyperbolicandellipticplanes General definition of the ζ-values for the points of the projective plane The ζ-values in the hyperbolic plane. The ζ-halfplane and the ζ-halfplane The hyperbolic determination of measure in the ζ-halfplane and on the ζ-halfsphere Remarksonsurfacesofconstantnegativecurvature Illustrations of the motions of the projective plane into itself by figures The elliptic plane and the ζ-plane resp. ζ-sphere Transferring the elliptic determination onto the ζ-plane and ζ-sphere The hyperbolic determination of measure in space and the associated motions Connection of the circle-relations with hyperbolic geometry. The rotationsubgroupsinhyperbolicspace Mapping of the hyperbolic space onto the ζ-halfplane Concludingremarkstotheintroduction Part I Foundations for the theory of the discontinuous groups of linear substitutions of one variable

2 XXX Contents 1 The discontinuity of groups with illustrations by simple examples Distinction between continuous and discontinuous substitution groups Distinction of properly and improperly discontinuous substitution groups Recapitulation and completion regarding the discontinuity domains ofcyclicgroups The groups of the regular solids and the regular divisions of the elliptic plane The division of the ζ-halfplane and the hyperbolic plane belonging to themodulargroup Introduction and extension of the Picard group with complex substitutioncoefficients The tetrahedral division of the ζ-halfsphere belonging to the Picard group The discontinuity domain and the generation of the Picard group RemarksonsubgroupsofthePicardgroup.Historicalmaterial The groups without infinitesimal substitutions and their normal discontinuity domains Theconceptofinfinitesimalsubstitutions The proper discontinuity of the groups without infinitesimal substitutions Introduction of the concept of the polygon and the polyhedron groups Introduction of the normal discontinuity domains of the projective planeforrotationgroups The vertices and edges of the normal polygons for principal circle groups.firstpart:thecornersintheinterioroftheellipse The vertices and edges of the normal polygons for principal circle groups. Second part: the vertices on and outside the ellipse The normal polyhedra in the hyperbolic space and their formation in theinteriorofthesphere Thenormalpolyhedraonandoutsidethesphere The behavior of the polygon groups on the surface of the sphere. First part:general Continuation: Special consideration of the groups with boundary curves The normal discontinuity domains for the groups consisting of substitutionsofthefirstandsecondkinds Carrying over the normal discontinuity domains onto the ζ-plane and into the ζ-space.historicalmaterial Further approaches to the geometrical theory of the properly discontinuous groups The allowed alteration of the discontinuity domains, in particular for principalcirclegroups Continuation: Allowed alteration of the discontinuity domains for polyhedral groups as well as non-principal circle polygon groups

3 XXXI 3.3 Definition of all groups without infinitesimal substitutions by suitable discontinuity domains. Effectuation in the principal circle case Continuation: Definition of the polyhedral groups by discontinuity domains Continuation: General definition of the polygon groups by suitable discontinuitydomains Classification of all groups without infinitesimal substitutions according to the form of the discontinuity domain and the regular divisionsarisingfromthese The generation of the groups and the relations subsisting between the generatingsubstitutions Continuation: The generators and their relations for polyhedron groupsaswellasforarbitrarypolygongroups Introduction of the closed resp. partially closed surfaces for polygon groupsofthefirstandsecondkinds The canonical discontinuity domains of the polygon groups Thecompositionofthepolygongroups Introduction of the homogeneous substitutions and groups The isomorphic splittability of polygon groups without secondary relations The homogeneous form of the primary relation between the V i,v ak,v bk 163 Part II The geometrical theory of the polygon groups of ζ-substitutions 1 Treatment of the rotation groups on the foundation of the normal discontinuity domains Disposaloftheellipticrotationgroups Thenormalhexagonoftheparabolicrotationgroups Relation of the normal hexagon of the parabolic rotation groups to the reductionofbinaryquadraticforms Theparabolicrotationgroupswithellipticsubstitutions Extension of the parabolic rotation groups by substitutions of the secondkind Continuation: Parabolic rotation groups of the second kind with ellipticsubstitutions The non-rotation groups with two boundary points Extension of the groups with two boundary points by substitutions of thesecondkind New explanations concerning the introduction of the normal polygons ofthehyperbolicrotationgroups Investigation concerning the cycles of accidental vertices of the normal polygons P Introduction of certain curves of the third order belonging to the substitution-triples V,V,V oftheaccidentalvertices

4 XXXII Contents 1.12 Additional remarks concerning the curves of third order of the triple V,V,V The domains Q belonging to the fixed polygon vertices. The reciprocity theoremofthenormalpolygon The species concept and the different types of normal polygons of the individualspecies The occurrence of special types of the normal polygons of the species (p,n) The alteration of the normal polygons for monodromy of the centers C The natural discontinuity domains of the hyperbolic rotation groups ofthefirstkind The canonical polygons and the moduli of the hyperbolic rotation groups Introductory remarks. The canonical polygons of the species (0, 3) The canonical polygons of the species (1,1) in their first form (as rectilinearquadrangles) The general form of the canonical polygons for the species (1,1) The double-n-angle of the species (0, n) anditstransformation Production of the canonical polygons of the species (0,n) fromthe double-n-angles The canonical polygons in the case of an arbitrary species (p,n) Continuation: Elimination of the convex angles possibly arising for the rectilinear canonical polygons of the species (p, n) Transformation theory of the canonical polygons of an arbitrary species (p,n) Continuation: The elementary transformations of the third and fourth kinds.finalresult The invariants of substitution pairs V 1,V Introduction of the moduli j 1, j 2, j 3 for the canonical polygons of the species(0,3) The system of the characteristic conditions for the moduli of the species(0,3).themanifoldofallgroups(0,3) The moduli and their characteristic conditions for the canonical polygonsofthespecies(1,1).manifoldofthegroups(1,1) Introduction of the moduli of the species (0, n). Composition considerationsanddeterminationsofsign Adjunction of the invariants j 123, j 234,... Relations for the moduli of the species (0, n) The setting up of further conditions valid for the moduli of the species (0,n) Systematic collection and completeness proof of the characteristic conditions for the moduli of the species (0, n) The families of group classes contained in the species (0, n): continuity considerationandexistenceproof The characteristic conditions for the moduli and the manifold of all groups of the species (p,n)

5 XXXIII 2.20 The transformation of the systems of moduli and the modular groups of the individual species (p, n) Special study of the modular transformations for the two species (0,4) and(1,1) Study of the circular-arc quadrangles without principal circle and remarks on other non-rotation groups Geometrical derivation of the seven types of circular-arc quadrangles Determination of the seven types of circular-arc quadrangles by their invariants Preparations for the investigation of the boundary curve for the quadrangleofthefirsttypewithfourangleszero Various approximative constructions of the boundary curve for the quadrangleofzeroangles The four kinds of points on the boundary curve G. Courseofthe boundarycurvewithrespecttoparabolicpoints Continuation: Course of the boundary curve at hyperbolic and loxodromicplaces Description of the quadrangle nets of the second through sixth types Further examples of non-rotation groups of the first and second kinds. 354 Part III Arithmetic methods of definition of properly discontinuous groups of ζ-substitutions 1 The rotation subgroups inside the Picard groups and the associated binary quadratic forms TheGaussianformsandthemodulargroup Introduction of the Dirichlet and Hermitian quadratic forms Geometrical interpretation of the Dirichlet and Hermitian forms Treatment of the equivalence problem for the definite Hermitian forms ReductiontheoryoftheDirichletforms ThetransformationoftheDirichletformsintothemselves ReductiontheoryoftheindefiniteHermitianforms ThereproducinggroupsoftheindefiniteHermitianforms The reproducing groups of the Hermitian forms belonging to the determinant D = The reproduction group of the Hermitian forms belonging to the determinant D = TheoryoftheGaussianformsinprojective-geometricalform TheprojectiveformofthePicardgroup Theory of the Hermitian and Dirichlet forms in projective-geometric form The reproducing groups of ternary and quaternary quadratic forms Approach to the groups to be investigated and proper discontinuity of these

6 XXXIV Contents 2.2 Equivalence and commensurability of the reproducing groups Γ f and Γ F Existence proof of the reproducing groups of the ternary forms f (z i ) ofbothkinds Existence proof of the reproducing groups of quaternary forms F (z i ) The occurrence of elliptic and parabolic substitutions in the groups Γ f and Γ F Historical remarks concerning ternary and quaternary forms ReportonSelling streatmentoftheternaryquadraticforms Arithmetic formation law of the ζ-groups Γ f of the indefinite forms f (z i ) New methods of construction of the discontinuity domain of the individual principal circle group Γ f Examples of reproducing groups of real forms f (z i ) Continuation: Groups [p, q, r ] incommensurable with the modular group Arithmetic formation law of the ζ-groups Γ f for complex ternary forms f (z i ) Examples of reproducing polyhedron groups Γ f Arithmetic formation laws for the reproducing groups of quaternary forms F (z i ) Example of a reproducing group Γ F A special kind of principal-circle and polyhedron groups with integral algebraic substitution coefficients Definition of the groups [p, q,r ] for arbitrary number fields Ω Various extensions of the groups [p, q, r ] Lemmasfromthetheoryofunits The discontinuity of the groups [p, q, r ] with real substitution coefficients The discontinuity of the groups [p, q, r ] with complex substitution coefficients Approach to the principal-circle groups of the character (0, 3) to be treated Discussion of the three conditions for proper discontinuity The groups [p, q,r ] belonging to the signatures (0,3;2,l 2,l 3 ) classified Proof of the identity of the groups (0,3;2,l 2,l 3 ) classified and the arithmetically defined group [p, q, r ] Approach to the principal circle groups of the character (1,1) to be treated The groups (1,1;l) with quadratic fields Ω for l = 4,5, The groups (1, 1; 2) and (1, 1; 3) with quadratic numbers fields Ω Commentaries 1 Commentary by Richard Borcherds on Elliptic Modular Functions

7 XXXV 2 Commentary by Jeremy Gray Commentary by William Harvey on Automorphic Functions Commentary by Barry Mazur Commentary by Series-Mumford-Wright Commentary by Domingo Toledo Commentaries by Other Mathematicians Index