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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.

MSC:

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