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Let $T_{g,n}$ be a compact orientable topological surface of genus $g$ with$n$ marked points and let $M(g,n)$ denote its mapping class group. We study theaction on the fundamental group of the surface induced by theself-homeomorphisms of the mapping class group by describing completely thecase of positive genus. In order to do so, we use a presentation for themapping class group with generators the Humphries generators and compute theirimages as elements of the outer automorphism group of the fundamental group,since the action is defined only up to conjugation.

The action of the Mapping Class Group on the fundamental group of the complement of a finite subset of a Riemann surface of positive genus