Groups Generated by two Parabolic Linear Fractional Transformations

Lyndon, R. C. ; Ullman, J. L.

Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1388-1403

Résumé

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

Détail

DOI : 10.4153/CJM-1969-153-1

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