Monodromy permutation ($\rho (\gamma )\in \mathrm{Sym}(p^{-1}(x_0))$)

Lifting a based loop in a covering space from every point of its finite fibre gives a permutation of that fibre. Composition of lifted loops yields the monodromy action of a covering space. If a closed geodesic of length $\ell$ has a cycle of length $m$ in its monodromy permutation, the corresponding lift closes after $m$ traversals and has length $m\ell$. For a primitive base geodesic, these cycles describe the primitive lift components.