Real quadratic irrational numbers and the group \(\langle x,y:x^2=y^6=1\rangle\). (Q1879039)

The modular group \(\text{PSL}(2,\mathbb{Z})\) acts on every real quadratic field by fractional linear transformations. The action of a subgroup generated by an element \(x\) of order 2 and an element \(y\) of order 6 is discussed. Here is a typical result: if \(\alpha\) is totally positive then \(y^k(\alpha)\) is totally negative for \(k=1,2,3,4,5\).