Split metacyclic actions on surfaces

Let \(\mathrm{Mod}(S_g)\) be the mapping class group of the closed orientable surface \(S_g\) of genus \(g\geq 2\). In this paper, we derive necessary and sufficient conditions under which two torsion elements in \(\mathrm{Mod}(S_g)\) will have conjugates that generate a finite split non-abelian metacyclic subgroup of \(\mathrm{Mod}(S_g)\). As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of \(\mathrm{Mod}(S_g)\) up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of \(S_g\). Moreover, for \(g \geq 5\), we show the existence of an infinite dihedral subgroup of \(\mathrm{Mod}(S_g)\) that is generated by an involution and a root of a bounding pair map of degree \(3\). Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of \(\mathrm{Mod}(S_3)\) and \(\mathrm{Mod}(S_5)\). We also describe nontrivial geometric realizations of some of these actions.