The slice determined by moduli equation \(xy=2z\) in the deformation space of once punctured tori (Q2565066)

Let \(A\), \(B\) be loxodromic elements of \(\text{PSL}(2,\mathbb{C})\), i.e. their traces do not lie on the closed interval \([-2,2]\). If one denotes by \(x\), \(y\), \(z\) the traces of \(A\), \(B\), \(AB\) respectively then the triple \((x,y,z)\in\mathbb{C}^3\) is called the moduli triple of the Möbius subgroup \(G=\langle A,B\rangle\). If \(\Omega(G)\) denotes the region of discontinuity of \(G\) and if \(\Omega(G)/G\) is happened to be a once punctured tori then the moduli triple \((x,y,z)\) of this \(G\) satisfies \[ x^2+ y^2+ z^2= xyz.\tag{\(*\)} \] The set \(T^*\) in \(\mathbb{C}^3\) consisting of triples \((x,y,z)\) satisfying \((*)\) is called a deformation space of once punctured tori. The present paper, as well as previous one, is devoted to determining other possible relations, which should satisfy the triple \((x,y,z)\in T^*\) to represent a moduli triple of once punctured tori.