Metacyclic actions on surfaces

Let [Formula: see text] be the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text]. In this paper, we derive necessary and sufficient conditions under which two torsion elements in [Formula: see text] will have conjugates that generate a finite metacyclic subgroup of [Formula: see text]. This yields a complete solution to the problem of liftability of periodic mapping classes under finite cyclic covers. As applications of the main result, we show that [Formula: see text] is a realizable upper bound on the order of a non-split metacyclic action on [Formula: see text] and this bound is realized by the action of a dicyclic group. Moreover, we give a complete characterization of the dicyclic subgroups of [Formula: see text] up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that every periodic mapping class in a non-split metacyclic subgroup of [Formula: see text] is reducible. We provide necessary and sufficient conditions under which a non-split metacyclic action on [Formula: see text] factors via a split metacyclic action. Finally, we provide a complete classification of the weak conjugacy classes of the finite non-split metacyclic subgroups of [Formula: see text] and [Formula: see text].