A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). 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.
Lowenthal, Franklin. Uniform Finite Generation of SU(2) and SL(2, R). Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 713-727. doi: 10.4153/CJM-1972-067-x
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author = {Lowenthal, Franklin},
title = {Uniform {Finite} {Generation} of {SU(2)} and {SL(2,} {R)}},
journal = {Canadian journal of mathematics},
pages = {713--727},
year = {1972},
volume = {24},
number = {4},
doi = {10.4153/CJM-1972-067-x},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1972-067-x/}
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[1] 1.
Lowenthal, F., Uniform finite generation of the rotation group, Rocky Mountain J. Math.1 (1971), 575–586. Google Scholar
[2] 2.
Lowenthal, F., Uniform finite generation of the isometry groups of Euclidean and non-Euclidean geometry, Can. J. Math.23 (1971), 364–373. Google Scholar
[3] 3.
Lowenthal, F., On subsemigroups of the projective group on the line, Can. J. Math.20 (1968), 1001–1011. Google Scholar
[4] 4.
Sternberg, S., Lectures on Differential Geometry (Prentice-Hall, Englewood Cliffs, N.J., 1964). Google Scholar
[5] 5.
Hidehiko, Yamabe, On an arcwise connected subgroup of a Lie group, Osaka J. Math.2 (1950), 13–14. Google Scholar
