Front Matter
0 Introduction. Developments concerning projective determinations of measure
  0.1 The projective determinations of measure in the plane and their division into kinds
  0.2 The motions belonging to a determination of measure and symmetric transformations of the plane into itself. The variable ζ in the parabolic case
  0.3 Setting up all collineations of the conic section z1z3 ?z22= 0 into itself. Behavior of the associated ζ
  0.4 The group of the “motion and symmetric transformations” for the hyperbolic and elliptic planes
  0.5 General definition of the ζ-values for the points of the projective plane
  0.6 The ζ-values in the hyperbolic plane. The ζ-halfplane and the ζ-halfplane
  0.7 The hyperbolic determination of measure in the ζ-halfplane and on the ζ-halfsphere
  0.8 Remarks on surfaces of constant negative curvature
  0.9 Illustrations of the motions of the projective plane into itself by figures
  0.10 The elliptic plane and the ζ-plane resp. ζ-sphere
  0.11 Transferring the elliptic determination onto the ζ-plane and ζ-sphere
  0.12 The hyperbolic determination of measure in space and the associated “motions”
  0.13 Connection of the circle-relations with hyperbolic geometry. The rotation subgroups in hyperbolic space
  0.14 Mapping of the hyperbolic space onto the ζ-halfplane
  0.15 Concluding remarks to the introduction
Part I Foundations for the theory of the discontinuous groups of linear substitutions of one variable
  1 The discontinuity of groupswith illustrations by simple examples
   1.1 Distinction between continuous and discontinuous substitution groups
   1.2 Distinction of properly and improperly discontinuous substitution groups
   1.3 Recapitulation and completion regarding the discontinuity domains of cyclic groups
   1.4 The groups of the regular solids and the regular divisions of the elliptic plane
   1.5 The division of the ζ-halfplane and the hyperbolic plane belonging to themodular group
   1.6 Introduction and extension of the Picard group with complex substitution coefficients
   1.7 The tetrahedral division of the ζ-halfsphere belonging to the Picard group
   1.8 The discontinuity domain and the generation of the Picard group
   1.9 Remarks on subgroups of the Picardgroup.Historicalmaterial
  2 The groups without infinitesimal substitutions and their normal discontinuity domains
   2.1 The concept of infinitesimal substitutions
   2.2 The proper discontinuity of the groups without infinitesimal substitutions
   2.3 Introduction of the concept of the polygon–and the polyhedron–groups
   2.4 Introduction of the normal discontinuity domains of the projective plane for rotation groups
   2.5 The vertices and edges of the normal polygons for principal circle groups. First part: the corners in the interior of the ellipse
   2.6 The vertices and edges of the normal polygons for principal circle groups. Second part: the vertices on and outside the ellipse
   2.7 The normal polyhedra in the hyperbolic space and their formation in the interior of the sphere
   2.8 The normal polyhedra on and outside the sphere
   2.9 The behavior of the polygon groups on the surface of the sphere. First part: General
   2.10 Continuation: Special consideration of the groups with boundary curves
   2.11 The normal discontinuity domains for the groups consisting of substitutions of the first and second kinds
   2.12 Carrying over the normal discontinuity domains onto the ζ-plane and into the ζ-space.Historical material
  3 Further approaches to the geometrical theory of the properly discontinuous groups
   3.1 The allowed alteration of the discontinuity domains, in particular for principal circle groups
   3.2 Continuation: Allowed alteration of the discontinuity domains for polyhedral groups as well as non-principal circle polygon groups
   3.3 Definition of all groups without infinitesimal substitutions by suitable discontinuity domains. Effectuation in the principal circle case
   3.4 Continuation: Definition of the polyhedral groups by discontinuity domains
   3.5 Continuation: General definition of the polygon groups by suitable discontinuity domains
   3.6 Classification of all groups without infinitesimal substitutions according to the form of the discontinuity domain and the regular divisions arising fromthese
   3.7 The generation of the groups and the relations subsisting between the generating substitutions
   3.8 Continuation: The generators and their relations for polyhedron groups aswell as for arbitrary polygon groups
   3.9 Introduction of the closed resp. partially closed surfaces for polygon groups of the first and second kinds
   3.10 The canonical discontinuity domains of the polygon groups
   3.11 The composition of the polygon groups
   3.12 Introduction of the homogeneous substitutions and groups
   3.13 The isomorphic splittability of polygon groups without secondary relations
   3.14 The homogeneous formof the primary relation between the Vi ,Vak ,Vbk
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
   1.1 Disposal of the elliptic rotation groups
   1.2 The normal hexagon of the parabolic rotation groups
   1.3 Relation of the normal hexagon of the parabolic rotation groups to the reduction of binary quadratic forms
   1.4 The parabolic rotation groupswith elliptic substitutions
   1.5 Extension of the parabolic rotation groups by substitutions of the second kind
   1.6 Continuation: Parabolic rotation groups of the second kind with elliptic substitutions
   1.7 The non-rotation groups with two boundary points
   1.8 Extension of the groups with two boundary points by substitutions of the second kind
   1.9 New explanations concerning the introduction of the normal polygons of the hyperbolic rotation groups
   1.10 Investigation concerning the cycles of accidental vertices of the normal polygons P0
   1.11 Introduction of certain curves of the third order belonging to the substitution-triples V,V ,V of the accidental vertices
   1.12 Additional remarks concerning the curves of third order of the triple V,V ,V
   1.13 The domainsQ belonging to the fixed polygon vertices. The reciprocity theoremof the normal polygon
   1.14 The species concept and the different types of normal polygons of the individual species
   1.15 The occurrence of special types of the normal polygons of the species (p,n)
   1.16 The alteration of the normal polygons formonodromy of the centers C0
   1.17 The “natural” discontinuity domains of the hyperbolic rotation groups of the first kind
  2 The canonical polygons and themoduli of the hyperbolic rotation groups
   2.1 Introductory remarks. The canonical polygons of the species (0,3)
   2.2 The canonical polygons of the species (1,1) in their first form (as rectilinear quadrangles)
   2.3 The general formof the canonical polygons for the species (1,1)
   2.4 The double-n-angle of the species (0,n) and its transformation
   2.5 Production of the canonical polygons of the species (0,n) fromthe double-n-angles
   2.6 The canonical polygons in the case of an arbitrary species (p,n)
   2.7 Continuation: Elimination of the convex angles possibly arising for the rectilinear canonical polygons of the species (p,n)
   2.8 Transformation theory of the canonical polygons of an arbitrary species (p,n)
   2.9 Continuation: The elementary transformations of the third and fourth kinds. Final result
   2.10 The invariants of substitution pairs V1,V2
   2.11 Introduction of the moduli j1, j2, j3 for the canonical polygons of the species (0,3)
   2.12 The system of the characteristic conditions for the moduli of the species (0,3). Themanifold of all groups (0,3)
   2.13 The moduli and their characteristic conditions for the canonical polygons of the species (1,1).manifold of the groups (1,1)
   2.14 Introduction of the moduli of the species (0,n). Composition considerations and determinations of sign
   2.15 Adjunction of the invariants j123, j234, . Relations for themoduli of the species (0,n)
   2.16 The setting up of further conditions valid for the moduli of the species (0,n)
   2.17 Systematic collection and completeness proof of the characteristic conditions for the moduli of the species (0,n)
   2.18 The families of group classes contained in the species (0,n): continuity consideration and existence proof
   2.19 The characteristic conditions for the moduli and the manifold of all groups of the species (p,n)
   2.20 The transformation of the systems of moduli and the modular groups of the individual species (p,n)
   2.21 Special study of the modular transformations for the two species (0,4) and (1,1)
  3 Study of the circular-arc quadrangles without principal circle and remarks on other non-rotation groups
   2.1 Geometrical derivation of the seven types of circular-arc quadrangles
   2.2 Determination of the seven types of circular-arc quadrangles by their invariants
   2.3 Preparations for the investigation of the boundary curve for the quadrangle of the first typewith four angles zero
   2.4 Various approximative constructions of the boundary curve for the quadrangle of zero angles
   2.5 The four kinds of points on the boundary curve G. Course of the boundary curvewith respect to parabolic points
   2.6 Continuation: Course of the boundary curve at hyperbolic and loxodromic places
   2.7 Description of the quadrangle nets of the second through sixth types
   2.8 Further examples of non-rotation groups of the first and second kinds
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
   1.1 TheGaussian forms and themodular group
   1.2 Introduction of the Dirichlet and Hermitian quadratic forms
   1.3 Geometrical interpretation of the Dirichlet and Hermitian forms
   1.4 Treatment of the equivalence problem for the definite Hermitian forms
   1.5 Reduction theory of theDirichlet forms
   1.6 The transformation of theDirichlet forms into themselves
   1.7 Reduction theory of the indefiniteHermitian forms
   1.8 The reproducing groups of the indefiniteHermitian forms
   1.9 The reproducing groups of the Hermitian forms belonging to the determinant D =5
   1.10 The reproduction group of the Hermitian forms belonging to the determinant D =7
   1.11 Theory of theGaussian forms in projective-geometrical form
   1.12 The projective formof the Picard group
   1.13 Theory of the Hermitian and Dirichlet forms in projective-geometric form
  2 The reproducing groups of ternary and quaternary quadratic forms
   2.1 Approach to the groups to be investigated and proper discontinuity of these
   2.2 Equivalence and commensurability of the reproducing groups Γf and ΓF
   2.3 Existence proof of the reproducing groups of the ternary forms f (zi ) of both kinds
   2.4 Existence proof of the reproducing groups of quaternary forms F(zi )
   2.5 The occurrence of elliptic and parabolic substitutions in the groups Γf and ΓF
   2.6 Historical remarks concerning ternary and quaternary forms
   2.7 Report on Selling’s treatment of the ternary quadratic forms
   2.8 Arithmetic formation law of the ζ-groups Γf of the indefinite forms f (zi )
   2.9 New methods of construction of the discontinuity domain of the individual principal circle group Γf
   2.10 Examples of reproducing groups of real forms f (zi)
   2.11 Continuation: Groups [p,q, r ] incommensurable with the modular group
   2.12 Arithmetic formation law of the ζ-groups Γf for complex ternary forms f (zi )
   2.13 Examples of reproducing polyhedron groups Γf
   2.14 Arithmetic formation laws for the reproducing groups of quaternary forms F(zi)
   2.15 Example of a reproducing group ΓF
  3 A special kind of principal-circle and polyhedron groups with integral algebraic substitution coefficients
   3.1 Definition of the groups [p,q, r ] for arbitrary number fields Ω
   3.2 Various extensions of the groups [p, q, r ]
   3.3 Lemmas fromthe theory of units
   3.4 The discontinuity of the groups [p, q, r ] with real substitution coefficients
   3.5 The discontinuity of the groups [p, q, r ] with complex substitution coefficients
   3.6 Approach to the principal-circle groups of the character (0, 3) to be treated
   3.7 Discussion of the three conditions for proper discontinuity
   3.8 The groups [p,q, r ] belonging to the signatures (0, 3; 2, l2, l3) classified
   3.9 Proof of the identity of the groups (0, 3; 2, l2, l3) classified and the arithmetically defined group [p, q, r ]
   3.10 Approach to the principal—circle groups of the character (1,1) to be treated
   3.11 The groups (1, 1; l ) with quadratic fields Ω for l = 4,5,6
   3.12 The groups (1, 1;2) and (1, 1;3) with quadratic numbers fields Ω
Commentaries
  1 Commentary by Richard Borcherds on EllipticModular Functions
  2 Commentary by Jeremy Gray
  3 Commentary byWilliam Harvey on Automorphic Functions
  4 Commentary by BarryMazur
  5 Commentary by Series-Mumford-Wright
  6 Commentary by Domingo Toledo
  7 Commentaries by OtherMathematicians
Index