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Groups Generated by two Parabolic Linear Fractional Transformations

Published online by Cambridge University Press:  20 November 2018

R. C. Lyndon  and
J. L. Ullman
R. C. Lyndon
Affiliation:
The University of Michigan, Ann Arbory Michigan
J. L. Ullman
Affiliation:
The University of Michigan, Ann Arbory Michigan
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We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the form

while Chang, Jennings, and Ree (4) took them in the form

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1388 - 1403
Copyright
Copyright © Canadian Mathematical Society 1969

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