Products of Axial Affinities and Products of Central Collineations

In order to obtain information about a mapping, it is often advantageous to factorize it into mappings of a special nature. We shall announce a number of results dealing with the factorization of affinities and projectivities into axial affinities and central collineations, respectively. These mappings are as simple as possible, since they leave all points of a hyperplane fixed. We shall distinguish different types of axial affinities such as shears, affine reflections, and affine hyperreflections, and of central collineations such as elations, projective reflections, and projective hyperreflections. We shall be interested in factorizations with a minimal number of factors for each mapping and also in finding upper bounds for the number of factors needed to express all mappings in a certain group.