Permutations and cubic graphs

J. L. Brenner and Roger Conant Lyndon

Vol. 104 (1983), No. 2, 285–315

DOI: 10.2140/pjm.1983.104.285

Abstract

In studying maximal nonparabolic subgroups of the modular group, B. H. Neumann and later C. Tretkoff were led to study pairs of permutations A and B of an infinite set Ω such that A² = B³ = 1 and that C = AB is transitive on Ω. Here we study such triples (Ω,A,B), but without the requirement that Ω be infinite. Our method is to associate with each such triple a graph G^∗(Ω,A,B). Such graphs have been used before, especially by Stothers and by Cori.

Our central result here is that the graphs under consideration are precisely those that can be obtained by attaching trees, in certain simple specified ways, to finite or infinite graphs equipped with a reduced path that traverses each edge exactly once in each direction.

Mathematical Subject Classification 2000

Primary: 20B05

Secondary: 05C75, 20F32

Milestones

Received: 17 March 1981

Revised: 18 March 1982

Published: 1 February 1983