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 Gal (Q ¯ / Q) in the case of cyclic covers of the projective line. [ABSTRACT FROM AUTHOR]