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Multicentral automorphisms in geometries of circles

Jakóbowski J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

2001

Vol. 49, no 4

417-432

We consider three types of geometries of circles (Moebius plane, Laguerre plane and Minkowski plane, cf. [4) with respect to so-called multicentral automorphisms. An automorphism [phi] of any geometry of circles is central if it has a fix point P and [phi] becomes a central collineation in the derived projective plane M(P). For any central automorphism [phi] we try to establish the whole set of points R such that [phi] becomes a central collineation in M(R.). Than [phi] is called multicentral if this set contains at least two points. Moreover, [phi] is proper if existing of a point [R is not equal to P], is not caused by the fact that [phi] is central in M(P). There is no proper multicentral automorphism in a Moebius plane. The most interesting proper multicentral automorphisms are involutorial mappings: double homotheties in Minkowski planes, and (sigma, tau)homologies in Laguerre planes. We give some examples.

Central automorphisms of Laguerre planes

Makowiecka H., Matraś A.

Demonstratio Mathematica

1998

Vol. 31, nr 4

753-763

Any automorphism of a Benz plane having at least one fixed point induces a collineation on the projective extension of the residual affine plane with reference to this point. When this collineation is a central automorphism , the initial automorphism is called the central automorphism (or central-axial automorphism, cf.([3]). In this paper we present an analytical description of central automorphisms of a miguelian Laguerre planes with the characteristic different from two. This description is applied to find transitive groups of homotheties and translations of types occuring in the classification theorems of R. Kleinewillinghofer ([2]). Some examples over an arbitrary commutative field are constructed, the other over the finite field Z3 and z5. It is interesting that two types of the Kleinewillinghofer classification ( [2] ) appear only as automorphism subgroups of finite plane of order three or five. This will give a clear characterization of these planes. Throughaut we assume that the characteristic of a plane is not equal to two.