Abstract
We consider cohomology sets and exact sequences of groups with involutions. In particular, we study congruence subgroups of type $\Gamma_0(m)$ which are acted by the group generated either by the map $z \mapsto (-1/mz)$ of the upper half plane or by the map $x \mapsto (1/mx)$ of the set of irrational real numbers.
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More From: Proceedings of the Japan Academy, Series A, Mathematical Sciences
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