Double cosets, rotations and isometric circles (Q2129475)

This paper is a short note on the properties of Möbius transformations. Algebraically, it presents a theorem that for any element \(g\) of \(\mathrm{SL}_2(C)\), there exists some positive real number \(\lambda\), and that the double coset \(UgU\) contains a hyperbolic transformation multiplying a positive real number \(\lambda\) where \(U\) denotes the unitary group \(U(2)\). The authors show that such \(\lambda\) is uniquely determined modulo its inverse as a stronger property. The proof of this exciting theorem is indeed elementary. Section 3 discusses the geometric meaning of this theorem. We characterize a Möbius transformation corresponding to an element of \(U\) as a transformation such that the chordal distance is preserved. If we assume that g is not unitary, then there exists a unique pair of antipodal points \(\{u, v\}\) such that \(\{g(u), g(v)\}\) is also antipodal. (If \(g\) is unitary, then every pair of antipodal points satisfies this property.) From this point of view, it follows that for non-unitary \(g\), \(g\) preserves the Apollonius circle family with two ends \(\{u, v\}\) as a set. It implies that \(g\) is a hyperbolic transformation with fixed points \(\{u,v\}\).