Multicentral automorphisms in geometries of circles (Q2777738)

The authors investigate multicentral automorphisms in Möbius, Laguerre and Minkowski planes. Such an automorphism has at least two fixed points and induces a central collineation in the projective extension of the derived affine plane at each of its fixed points.NEWLINENEWLINENEWLINEThe authors start from a central automorphism with fixed point \(P\) and analyse all configurations for the centre and axis of the induced collineation of the derived projective plane \(\overline{{\mathbb M}(P)}\) at \(P\) taking account of whether the centre is an affine point or on the ideal line and whether the axis comes from a circle, a generator or is the ideal line. For each other fixed point \(R\neq P\) they then determine whether or not one necessarily obtains a central collineation in \(\overline{{\mathbb M}(R)}\). This process gives rise to an awkward definition of a proper multicentral automorphism, that is, a type of central automorphism for which it cannot be generally infered that one obtains an induced central collineation in the derived projective plane at some other fixed point.NEWLINENEWLINENEWLINEThere are no proper multicentral automorphisms in Möbius planes and only one such kind in Minkowski planes. The latter are the involutory double homotheties determined by four points of a rectangle whose sides are formed by generators. (If one has an induced central collineation at one of the points in the rectangle with centre the opposite point one also has a central collineation at this opposite point but there are Minkowski planes and automorphisms for which the induced collineations at the adjacent points in the rectangle are central collineations and there are example for which they are not. Hence one obtains a proper multicentral automorphism in the former case.)NEWLINENEWLINENEWLINEFor Laguerre planes there are two types of proper multicentral automorphisms, those that fix at least two points on a generator and induce translations at each of them, and those that fix all the points of two generators and induce homologies at each of them.NEWLINENEWLINENEWLINEThe authors provide examples for each of their three types of proper multicentral automorphisms.