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It is know that any Neumann subgroup of the modular group is maximalnon-parabolic. The question arises as to whether these are the only maximalnon-parabolic subgroups. The wild class of maximal, non-parabolic, not Neumannsubgroups of the modular group was constructed by Brenner and Lyndon. The newconstruction of such a class is presented. Those groups are obtained assubgroups of elements of positive determinant of any Neumann subgroup of theextended modular group (the notion which we introduce in the paper) and in thissens they are as "close" as possible to Neumann subgroups of the modular group.

Neumann property in the extended modular group and maximal nonparabolic subgroups of the modular group