Action of a two generator group on a real quadratic field. (English) Zbl 1113.11058

Summary: We are interested in studying an action of the group \(G= \langle u,v: u^3= v^3=1\rangle\), where \(u\) and \(v\) are linear fractional transformations \(z\to \frac{z-1}{z}\) and \(z\to \frac{-1}{z+1}\) respectively on \(\mathbb Q(\sqrt{n})\) by using coset diagrams.
For a fixed non-square positive integer \(n\), if the real quadratic irrational number \(\alpha= \frac{a+\sqrt{n}}{c}\) and its algebraic conjugate \(\overline{\alpha}= \frac{a-\sqrt{n}}{c}\), have different signs, then \(\alpha\) is called an ambiguous number. They play an important role in classifying the orbits of \(G\) on \(\mathbb Q(\sqrt{n})\). In the action of \(G\) on \(\mathbb Q(\sqrt{n})\), \(\text{Stab}_\alpha(G)\) are the only non-trivial stabilizers and in the orbit \(\alpha G\); there is only one (up to isomorphism). We have also classified all the ambiguous numbers in the orbit.


11R11 Quadratic extensions
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11F06 Structure of modular groups and generalizations; arithmetic groups