Fenchel-Nielsen coordinates of the branch loci of cyclic actions

Let $S_g$ be a closed, connected, and oriented smooth surface of genus $g\ge 2$. Let the mapping class group of $S_g$ be denoted by $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ and the Teichmüller space of $S_g$ by $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(S_g)$. It is known that $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ acts by isometries on $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(S_g)$ with respect to the Weil-Petersson metric. In this paper, we develop algorithms to describe the Fenchel-Nielsen coordinates of fixed points of the actions of certain finite cyclic subgroups of $\mathrm{M}\mathrm{o}\mathrm{d}(S_g)$ on $\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{h}(S_g)$. As applications of these algorithms, we compute the Fenchel-Nielsen coordinates of the fixed points of three cyclic subgroups of orders $10$, $8$, and $4$, in $\mathrm{M}\mathrm{o}\mathrm{d}(S_2)$.