The Torelli geometry and its applications

Details The Torelli geometry and its applications The Torelli geometry and its applications

2003-11-10

arxiv.org/abs/math/0311123v1

Mathematics

Geometric Topology

Research announcement, 11 pages

Scientific paper

For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometry is induced by a diffeomorphism of the surface in question. We also provide an intrinsic algebraic characterization of some natural elements of the Torelli group and of some geometric relations between them. When combined, these results allow us to compute the automorphism group, the outer automorphism group and the abstract commensurator of the Torelli group for surfaces of genus at least 5.

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