Hostname: page-component-84b7d79bbc-g5fl4 Total loading time: 0 Render date: 2024-08-02T07:19:07.327Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Lowenthal, F., Uniform finite generation of the rotation group, Rocky Mountain Journal of Mathematics (to appear).Google Scholar
2. Lowenthal, F., Uniform finite generation of the affine group, Pacific J. Math, (submitted for publication).Google Scholar
3. Lowenthal, F., On generating subgroups of the Moebius group by pairs of infinitesimal transformations, Pacific J. Math. 26 (1968), 141–147.Google Scholar
4. Lowenthal, F., Qn subsemigroups of the projective group on the line, Can. J. Math. 20 (1968), 1001–1011.Google Scholar
5. Sternberg, S., Lectures on differential geometry, pp. 45–55 (Prentice-Hall, Englewood Cliffs, N.J., 1964).Google Scholar