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On Subsemigroups of the Projective Group on the Line

Published online by Cambridge University Press:  20 November 2018

Franklin Lowenthal
Franklin Lowenthal*
Affiliation:
University of Oregon, Eugene, Oregon
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The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1001 - 1011
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

Theorems 1 and 1* and one case of Theorem 2 are part of the author's doctoral dissertation, written under the guidance of the late Professor Charles Loewner. Additional research was supported by a grant from the University of Oregon.

References

1. In (2), the solution can be put into the form y = (ax + b)/(cx + d), where a, b, c, and d are all real and ad - be 0.

2. Assume, for definiteness, that the sink interval is just (zi, W\) which can be achieved by inner automorphism if necessary.

3. Observe that this is clearly a product of length 3. As the order of applying transformations always begins at the right, parentheses will be omitted subsequently.

4. The parameters t, s will be used from now on.

5. In fact, ϒ = V(T2), ϒ ≧ l , but this is not needed. It shows that ϒ (τ)is monotonically increasing, a fact which subsequently is not used.

6. If ॉ is elliptic, and ॉ and n have a common root, then they generate the same subgroup.