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

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 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 364 - 373
Copyright
Copyright © Canadian Mathematical Society 1971

References

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3. Lowenthal, F., On generating subgroups of the Moebius group by pairs of infinitesimal transformations, Pacific J. Math. 26 (1968), 141–147.Google Scholar
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5. Sternberg, S., Lectures on differential geometry, pp. 45–55 (Prentice-Hall, Englewood Cliffs, N.J., 1964).Google Scholar