Uniform Finite Generation of the Isometry Groups of Euclidean and Non-Euclidean Geometry

Abstract

A connected Lie group H is generated by a pair of oneparameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups. If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n; otherwise define it as infinity.For the isometry group of the spherical geometry, or equivalently for the rotation group SO(3), the order of generation is always finite.