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Uniform Finite Generation of SU(2) and SL(2, R)

Published online by Cambridge University Press:  20 November 2018

Franklin Lowenthal
Franklin Lowenthal*
Affiliation:
University of Oregon, Eugene, Oregon
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 713 - 727
Copyright
Copyright © Canadian Mathematical Society 1972

References

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5. Hidehiko, Yamabe, On an arcwise connected subgroup of a Lie group, Osaka J. Math. 2 (1950), 13–14.Google Scholar