Group Actions on cyclic covers of the projective line

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\text{Q}}\mathrm{/}\text{Q})$ in the case of cyclic covers of the projective line.