The structure of the Torelli group. I: A finite set of generators for \({\mathcal I}\).

*(English)*Zbl 0549.57006Let \({\mathfrak M}={\mathfrak M}_{g,n}\) be the mapping class group of a compact orientable surface \(M=M_{g,n}\) of genus g with n boundary components \((=diffeomorphisms\) modulo isotopy, also called the Teichmüller modular group). For \(n=0\) or 1, let \(T=T_{g,n}\) be the subgroup of \({\mathfrak M}_{g,n}\) which acts trivially on \(H_ 1(M;{\mathbb{Z}})\); this is called the Torelli group.

In the present paper, answering a well-known question, it is shown that \(T_{g,0}\) and \(T_{g,1}\) are finitely generated for \(g\geq 3\). Powell showed that \(T_{g,0}\) is generated by the following infinite set of elements: Dehn-twists along bounding simple closed curves; opposite Dehn- twists along a pair of disjoint homologous simple closed curves. The author of the present paper constructs a finite set of generators of the second kind for the Torelli group \(T_{g,1}\) and \(T_{g,0}\), \(g\geq 3\). It has recently been shown by McCullough and Miller that the Torelli group \(T_{2,0}\) (and \(T_{2,1})\) is not finitely generated (note that for \(g=2\) only generators of the first kind occur).

In the present paper, answering a well-known question, it is shown that \(T_{g,0}\) and \(T_{g,1}\) are finitely generated for \(g\geq 3\). Powell showed that \(T_{g,0}\) is generated by the following infinite set of elements: Dehn-twists along bounding simple closed curves; opposite Dehn- twists along a pair of disjoint homologous simple closed curves. The author of the present paper constructs a finite set of generators of the second kind for the Torelli group \(T_{g,1}\) and \(T_{g,0}\), \(g\geq 3\). It has recently been shown by McCullough and Miller that the Torelli group \(T_{2,0}\) (and \(T_{2,1})\) is not finitely generated (note that for \(g=2\) only generators of the first kind occur).

Reviewer: B.Zimmermann

##### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

20F05 | Generators, relations, and presentations of groups |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

57R50 | Differential topological aspects of diffeomorphisms |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |