2003/11/10 by Benson Farb, Farb, Benson, Nikolai V. Ivanov +1
Mathematics · #20F38 #30F60 #32G15 (Primary) #57M07 #57M99 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT #msc:20F38 #msc:30F60 #msc:32G15 #msc:57M07 #msc:57M99
paper · pdf · doi:10.48550/arxiv.math/0311123
Research announcement, 11 pages
arxiv created 2003/11/10 · arxiv updated 2009/12/01
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.