\(4\)-punctured spheres (Q1346103)

If \((g;e_1, \dots, e_r; m)\) is the signature of a Fuchsian group of the first kind, let \({\mathcal F}(g; e_1, \dots, e_r; m)\) denote the set of all Fuchsian groups of that signature. The author obtains very precise information on the set \({\mathcal F}(0; -; 4)\) i.e. the monodromy groups of (hyperbolic) 4-punctured spheres. More precisely, if \(I(G)\) denotes the subgroup generated by all involutions in \(G\), and \(G^+\), the normaliser in \(\text{PSL} (2, \mathbb{R})\), then the map \(G \to I(G^+)\) is a bijection from \({\mathcal F} (0; -; 4)\) to \({\mathcal F} (0;2,2,2;1)\) and three distinct possibilities for \(G\) arise, determined by the nature of \(G^+/G\), being a 4-group, a dihedral group of order 8, or isomorphic to \(\text{PSL} (2,3)\). The methods are direct, but cunning calculations. The braid group on 4- strings is related to the mapping class group of the 4-punctured sphere, and some of the calculations are shown to be motivated by an invariance under certain braid group actions.