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The structure of the Torelli group. I: A finite set of generators for $${\mathcal I}$$. (English) Zbl 0549.57006
Let $${\mathfrak M}={\mathfrak M}_{g,n}$$ 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).
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)
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