Maximal Nonparabolic Subgroups of the Modular Group
نویسنده
چکیده
The elliptic elements of M, each with two conjugate complex fixed points, are precisely the conjugates of nontrivial powers of A and B. The parabolic elements, each with a single real fixed point, are precisely the conjugates of nontrivial powers of C = AB: z ~ z + 1. The remaining nontrivial elements of M are hyperbolic, each with two real fixed points. A subgroup S of M is torsionfree (and therefore by the Kurosh St~bgroup Theorem a free group) if and only if it contains no elliptic elements. A subgroup S of M is called nonparabolic if it contains no parabolic elements. N eumann [9] showed that, if T is the infinite cyclic group generated by a conjugate of C, and S is a complement to T in M in the sense that Sc~T=I and S T = M , then S is a ~ maximal nonparabolic subgroup of M. We call such subgroups Neumann subgroups. Magnus [8] raised the question whether all maximal nonparabolic subgroups of the modular group are Neumann subgroups. We show that this is not the case. We show that the kernel N of the obvious map from M onto the planar crystallographic group
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