Uniform Finite Generation of Threedimensional Linear Lie Groups

Abstract

A connected Lie group G is generated by one-parameter subgroups exp(tX1), … , exp(tXk) if every element of G can be written as a finite product of elements chosen from these subgroups. This happens just in case the Lie algebra of G is generated by the corresponding infinitesimal transformations X1, … , Xk ; indeed the set of all such finite products is an arcwise connected subgroup of G, and hence a Lie subgroup by Yamabe's theorem [9]. If there is a positive integer n such that every element of G possesses such a representation of length at most n, G is said to be uniformly finitely generated by the one-parameter subgroups.