Coset graphs
R. C. Lyndon · Cambridge University Press eBooks · 1987
1. This is a report on a series of joint papers with Joel Brenner [BL 1-6]. Our work was prompted by a 1933 paper of Bernhard Neumann [Nl] arising, I am told, from a problem in the foundations of geometry, or, more immediately, by Carol Tretkoff's 1975 sequel [Tl] to Neumann's paper. Neumann was led to study maximal nonparabolic subgroups of the modular group. The modular group M has a presentation M = (a,b : a 2 = b 3 = 1). It is well known to be isomorphic to the group PSL(2,Z) and to have a representation on the extended complex plane ℂ* given by The element c = ab : z + z + 1 is parabolic in the sense of having only a single fixed point in ℂ*, and it is easily shown that the parabolic elements of M are exactly the conjugates of nontrivial powers of c. A subgroup P of M is a parabolic group if all its nontrivial elements are parabolic, and a subgroup S is nonparabolic if it contains no parabolic element. The maximal parabolic sugroups are exactly the conjugates of the infinite oyclic group C generated by c. Neumann observed that if P is a maximal parabolic subgroup and if S is a complement to P in the sense that S ∩ P = 1 and SP = M, then S is a maximal nonparabolic subgroup.