Geometry/Groups

Geometric group theory

Modern geometry is expressed with group theory. The following expression of the role of groups in geometry was given by w:Felix Klein:

geometric properties of any figures must be expressible in formulas which are not changed when one changes the coordinate system, i.e. when one subjectas all the points of the figure simultaneously to one of our transformations, and , conversely, any formula which, in this sense, is invariant under the group of these transformations must represent a geometric property. As simplest examples, which all of you know, let me remind you of the expression for distance or for angle, in the figure of two points or of two lines.

This algebraic attitude to geometry is called the Erlangen Program.

Formal statement

Let X be a set of points and S a set of subsets of X. For example, s in S may represent a line or a circle or some other characteristic feature of X. Consider A a set of axioms about X and S. Finally, let P be a proposition expressing a feature of elements of S.

Suppose b is a bijection of X with itself. The proposition Pb is obtained from P by exchanging all mentions of elements of S in P by their images under b. Now consider the set of all bijections that respect the property represented by proposition P : $G=\{b:P\equiv Pb\}.$

If c is in G, then $P\equiv Pb\equiv Pc\implies P\equiv Pbc,$ so that bc is in G. Also the inverse of b is in G, so G is a group.

Definition: The geometry $G=(X,\ S,\ A,\ P)$ is the transformation group determined by property P.

First consider the plane X = R² with the property P given by the distance between two points:

$d((w,x),(y,z))={\sqrt {(w-y)^{2}+(x-z)^{2}}}.$ Distance is invariant under rotation, translation, or reflection in a line. Therefore G is the group generated by these transformations: the Euclidean group of the plane.

In space X = R³, the distance between a pair of points represents P and generates the corresponding Euclidean space group. Consider a screw displacement given by a rotation about an axis and a translation parallel to that axis. According to a theorem in kinematics (attributed to Mozzi and Chasles), any motion in the Euclidean space group may be represented as a screw displacement.

Affine geometry

Returning to the plane X = R², let P be the property of parallel lines. Thus b is in G when two parallel lines are taken by b to another pair of parallels. Then G is the affine group, which contains the Euclidean group but also includes squeeze mappings that transform a square to a rectangle of the same area as the square. This group has found application in flat-space cosmology where light rays trace lines through spacetime. In fact, a squeeze mapping in the affine group corresponds to a leap from a frame of reference determined by one velocity to one with another velocity.

The conversion of geometry, from properties of configurations of points, lines and other features of geometric space, to group theory, was accomplished by Felix Klein. He was put on the path to this conversion by Arthur Cayley's piece "On the theory of distance" (1859), which obtained a metric space known as the elliptic plane from the real projective plane by use of logarithm of crossratio. Klein ran with the idea, demonstrating the models for the hyperbolic plane, and he established non-Euclidean geometry as a well-founded branch of mathematics. He articulated a philosophy of geometry via group theory, where when property P implies Q, then G_P is contained in G_Q, as in the case of Euclidean and affine geometries.

The interior of the unit disk $D=\{z:|z|<1\}$ represents the hyperbolic plane in one model. Any circle intersecting D orthogonally on the boundary represents a line in this hyperbolic plane. Evidently, for a point in D but not on a given line, there are many lines through the point that do not intersect the given line in D. This model, with its expression of motion by Mobius transformations leaving D stable, gave evidence of the consistency of the earlier theoretical hyperbolic plane described by Bolyai and Lobachevskii. Thus geometry was expanded beyond Euclid to the non-Euclidean, and the classical field became a branch of group theory.

It was in 1872 at Erlangen, Germany, that Felix Klein first articulated his group-philosophy of geometry. Since that time the subsequent movement has been labeled the w:Erlangen program.

Conformal geometry

A conformal transformation is one that preserves angles: the angle between two intersecting lines is the same before and after the transformation. Then the inverse transformation also preserves angles. The set of conformal transformations, combined by composing one after the other, forms a group under composition, with the identity mapping as the group identity.

Translations of a plane are obviously conformal, so attention is directed to transformations that leave a point invariant. This point is taken as the origin (0,0) of a coordinate plane $\{(x,\ y):x,y\in \mathbb {R} \}.$

Conformal geometry (X, S, A, P) corresponds to X as the coordinate plane, S the set of angles between lines in the plane, and P the requirement of equality of angle measure before and after transformation. The axioms A for the angles can refer to circular angle, hyperbolic angle, or slope. In the case of slope, the conformal geometry is given by the group of shear mappings. In the case of hyperbolic angle, the conformal geometry is given by the group of squeeze mappings. Naturally, when the axioms specify the circular angle, then the group of rotations are the conformal group.

Ideally all the axioms A could be gathered together. Indeed, when angle is defined by means of area, then the property P can be specified as preserving area. The various axioms for the various angles refer to bounding arcs of sectors associated with the angle in the theory of unified angles.

Linear algebra forms the natural context for this topic in group theory. The transformations are expressed with matrices. Every matrix has a number called the determinant, and when this number is 1 or −1, the matrix expresses a transformation that preserves areas.