  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.
