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Dynamical types and conjugacy classes of centralizers in groups. (English) Zbl 1181.22022
Summary: The transformations in classical geometries often come equipped with natural spatial and numerical invariants. Moreover, although they are infinitely many transformations, their “dynamical types” are finite in number. While not attempting to define the notion of a dynamical type, we attempt to explain these phenomena by relating them to the centralizer-conjugacy classes, or $z$-classes, of elements in the automorphism groups of these geometries. We prove a fibration theorem 2.1 for an orbit class in a general group-action, and specialize it to the $z$-classes. It describes two set-theoretic fibrations of each orbit class, in particular a $z$-class. The base and the fiber in these fibrations of a $z$-class provide a partial explanation of the spatial and numerical invariants. The possible finiteness of dynamical types is related to the possible finiteness of $z$-classes. We bring out the implicit role played by the field of real numbers in the assertion of finiteness of dynamical types of transformations in classical geometries. The analysis shows that the finiteness of dynamical types is expected to hold in the analogues of classical geomeries defined over other local fields. Two examples, namely i) the plane Euclidean geometry, and ii) the semisimple operators on a vector space over an arbitrary field, are worked out in detail to illustrate the general ideas.

22F50Groups as automorphisms of other structures
37A15General groups of measure-preserving transformation
14L35Classical groups (geometric aspects)
51N30Geometry of classical groups
51M10Hyperbolic and elliptic geometries (general) and generalizations
51M05Euclidean geometries (general) and generalizations
20G15Linear algebraic groups over arbitrary fields