Dynamical Types of Transformations in Classical Geometries

A classical theorem of Euler says that a motion of a solid body having a fixed point is a rotation around a fixed axis. Similarly in 2-dimensional Euclidean geometry every orientation-preserving isometry is a rotation though some angle around a fixed point, or else it is a translation. Similarly in 2-dimensional hyperbolic geometry every orientation-preserving isometry is a rotation though some angle around a fixed point, or a parabolic fixing exactly one point on the ideal boundary, or a hyperbolic fixing exactly two points on the ideal boundary. There are two remarkable features in these examples: 1) Each transformation is characterized by a spatial invariant and a numerical invariant taking values in Abelian groups. 2) Although the groups are infinite, there are only finitely many "dynamical types". A cursory examination of other classical geometries, such as higher-dimensional Riemannian geometries of constant curvature, projective geometry, Mobius geometry ... shows that these features continue to persist. A natural problem is to account for these features in a systematic way.

We shall account for the first feature by proving a general result on group actions. It generalizes the analysis initiated by H. Weyl in his study of compact Lie groups. When one applies it to a group acting on itself by conjugation, (or a Lie group acting on its Lie algebra by the adjoint action), one substantially recovers the observation regarding spatial and numerical invariants.

Accounting for the second feature is more subtle. It crucially depends on the fact that the groups, or the corresponding geometries are defined over real numbers. For geometries defined over fields with richer arithmetic (such as the field of rational numbers), the finiteness of "dynamical types" does not hold. There arise new arithmetic invariants, depending on allowable field extensions. A general problem is to understand the internal structure of k-points of an algebraic groups defined over k. In the second part of this talk we shall analyze the case of GL(n) defined over an arbitrary field. It may have some implication of our teaching of linear algebra.