DEFINING RELATIONS FOR REFLECTIONS. I OLEG VIRO

Abstract. An idea to present a classical Lie group of posi-tive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geome-tries admit elegant and useful presentations by generators and relations. They are closely related to geometry and allow to make fast and efficient calculations in the groups. In this paper simple presentations of the isometry groups of Euclidean plane, 2-sphere, the real projective plane and groups SO(3) and O(n) are introduced. 1. Isometries of the Euclidean plane 1.1. Well-known facts. See, e.g., [2]. Classification. Any isometry of the Euclidean plane is either the identity id, or a reflection in a line, or a translation, or a rotation about a point, or a glide reflection (composition of a reflection in a line and a translation along the same line). Generators. Any isometry is a composition of at most three reflections in lines. Compositions of refections. The composition Rm ◦ Rl of reflections Rl and Rm in lines l and m, respectively, is • the identity if l = m; • a translation if l ‖ m; • a rotation if l is transverse to m. If l ‖ m, then the translation Rm◦Rl moves any point by a distance twice greater than the distance between the lines in the direction perpendicular to these lines. If l is transverse to m, then the composition Rm ◦Rl is a rotation about the intersection point m ∩ l by the angle twice greater than the angle between the lines. See figure 1. 1 a