Le groupe d'automorphismes du groupe modulaire (Q1069511)

We give a new and simple proof of the fact that the full mapping class group \(M^*_ g\) of a closed oriented surface of genus \(g\geq 3\) is complete (this is known as Ivanov's theorem). In studying the action of an automorphism on finite subgroups of \(M^*_ g\), one remarks that hyperelliptic involutions are mapped onto themselves. The argument also uses Dyer's and Grossman's result asserting that the outer group of automorphisms of the braid group \(B_ n\) is isomorphic to \({\mathbb{Z}}_ 2\). The proof extends to the case of surfaces with one puncture. Unfortunately, our method doesn't prove the theorem of Ivanov in the case of surfaces with finitely many punctures.