Klein's Vision

Klein's Vision and Semantic Unfolding in the Geometric Conception of Potential Theory: Uniformization of Riemann Surfaces via the Non-Commutative Free Group in Two Generators

Felix Klein is famous for his Erlangen Program, which appeared in 1872 in the context of his professorial dissertation at the homonymous university. The central conceptual aspect of Klein’s Erlangen program is expressed by the thesis that the objective content of a geometric theory is captured by the group of transformations of a space. The crucial insight of Klein's program is that transformation groups constitute an algebraic encoding of a criterion of equivalence for geometric objects.

Principle of Transference: Isomorphism

Klein's thesis is expressed as the principle of transference, or principle of isomorphism, induced by a transitive group action on a space. A transfer of structure is taking place by means of an isomorphism providing different equivalent models of the same geometric theory. Philosophically, underneath there lies an Aristotelian conception of space, according to which space is conceived as being matter without form. The form is being enacted by the action of a concrete transformation group. Still, more important, the space itself may be considered as the quotient of the transformation group over a closed subgroup of the former. A change in algebraic form, or else, a change of transformation group signifies a change in geometry, in the sense that the equivalence criterion encoded in the group action is altered. Thus, moving from a group to a larger one amounts to a change in the resolution unit of figures, expressed as a relaxation of the geometric equivalence criterion involved in the procedure.

Relation with Complex Function Theory

The Erlangen program should not be considered independently from the deep influence that Riemann's geometric complex function theory had on Klein's thinking. In 1882 Klein wrote down an explanatory presentation of the theory of Riemann surfaces, entitled "On Riemann's theory of algebraic functions and their integrals''. In this monograph, he preserves the original physical conception of Riemann's function theory based on the notions of potential, flow, streamlines and conformal transformation, subsumed by Riemann mainly under the term ``Dirichlet's principle''. The pressing problem in this respect is what he calls ``Riemann's multiform correspondence''.

Grenzkreis Theorem

Every Riemann surface of genus greater than one can be represented uniquely by an invariant potential function not having any branch points that is defined on a disk bounded by a circle. The images of the Riemann surface on the sphere obtained under analytic continuation of this potential function swells out the region of a disk bounded by a circle, which in turn, constitutes the boundary that they indefinitely approach without overlapping. More concretely, the images of the Riemann surface on the sphere is a polygon made of circular arcs that are all perpendicular to a common circle, i.e. the boundary circle of the disk region on the sphere.

Klein-Poincare Relation

A discontinuous group action on a disk leading to the effectuation of some circular arc polygon within the region of this disk corresponds uniquely to a respective Riemann surface of genus bigger than one. The above is in full concordance with Poincare's viewpoint on the notion of Riemann surfaces. More precisely, Poincare's semantic unfolding in this setting is based on the fact that a Fuchsian group used to set up a certain tessellation of the hyperbolic disk by hyperbolic polygons is isomorphic to the fundamental group of the quotient Riemann surface made with that same polygon by isometric side pairings. What Klein missed in comparison to Poincare is the metrical manifestation and realization of his "Grenzkreis theorem''.

Erlangen Perspective on Homographies

Mobius transformations, that is, homographies are the symmetries of patterns on the Riemann sphere. They are classified into three types as loxodromic, parabolic, or elliptic. Loxodromic transformations have two fixed points, one of which may be physically thought of as attracting whereas the other one as repelling, and are conjugate to scaling by complex numbers except for scaling by unit complex numbers. Those ones whose multiplier is a positive real number are also called hyperbolic transformations. Parabolic transformations have one fixed point and are conjugate to parallel translations. Elliptic transformations have two fixed points and are conjugate to rotations.

Shottky Model and Unfolding Action of the Free Group

Two non-commuting loxodromic transformations acting jointly on the constellation of four non-overlapping disks, and without imposing any further relations, generate the free non-commutative group in two generators, called in this context the Schottky group. Repeated application of all the four transformations, corresponding to the actions of the generators and their inverses, leads to an action of this group on the constellation of four non-overlapping disks that is characterized by a repetitive pattern at all different levels of magnification. E

Each of the involved disks contains three smaller disks, each of which in turn contains three smaller disks, and so on ad infinitum. In this manner, disks within disks are nesting down leaving invariant at the limit a type of Cantorian dust, i.e. leaving invariant those points that belong to the disks at every single level of the process.

We may think of a single particle of dust at the limiting end point of each infinite chain of disks nested within each other by the action of the Schottky group. In this sense, the limit-set of this group action consists of these particles of dust, and hence, this limit-set is the invariant pattern under the action of the free group in the Schottky setting identified with the closure of all attracting and repelling points.

Galois Action of the Free Group

The Schottky is realized by its free, and thus autogenetic, action on a 4-valent tree, characterized by a boundless exponential schema of growth. In turn, this 4-valent tree being simply connected and amenable to the metric structure of its natural path metric qualifies as a geometric space whose group of symmetries is the free group in two non-commuting generators.

Taking into account that the action of the Schottky group on this 4-valent tree is free and transitive, it qualifies as a Galois action. In this manner, the Schottky group is realized as a group of covering automorphisms of a 4-valent tree in the role of the latter as a universal covering space annihilating the fundamental group of its quotient by the action of the free group.

In the present setting, the action of the Schottky group as a free group generated by two non-commuting loxodromic transformations which act jointly on a constellation of four non-overlapping disks on the Riemann sphere may be comprehended by means of the concomitant 4-valent tree. Equivalently, this 4-valent tree is able to record the associated pattern of nesting disks within disks under the action of the free group, by explicating graphically the organization pattern of nested disks at different levels of constellatory unfolding.

Free Group Uniformization Action and Koebe's Retrosection Theorem

Every compact Riemann surface can be represented via the uniformizing action of the free group on "g" generators where "g" is the genus of the Riemann surface. In particular, it can be realized as the quotient of the domain of discontinuity of this unfolding free group action modulo the free group itself. Not only this, but additionally, the image of the "2g" boundary circles on the Riemann sphere under the quotient morphism may be identified with "g+g" oriented loops and their inverses in the fundamental group of this Riemann surface.