Metacyclic actions on surfaces

Let $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\ge 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ will have conjugates that generate a finite metacyclic subgroup of $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$. 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 $4g$ is a realizable upper bound on the order of a non-split metacyclic action on $S_g$ and this bound is realized by the action of a dicyclic group. Moreover, we give a complete characterization of the dicyclic subgroups of $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ 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 $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ is reducible. We provide necessary and sufficient conditions under which a non-split metacyclic action on $S_g$ 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 $\mathrm{M}\mathrm{o}\mathrm{d}(S_{10})$ and $\mathrm{M}\mathrm{o}\mathrm{d}(S_{11})$.