Torelli buildings and their automorphisms

It is well known that each compact orientable surface R gives rise to a simplicial complex C(R) playing in the theory of Teichmüller modular groups a role similar to the role of Tits buildings in theories of algebraic and arithmetic groups. In this paper we introduce, for each closed orientable surface S, an analogue of Tits buildings more suitable for investigation of the Torelli group I(S) of S. It is a simplicial complex with some additional structure. We denote this complex with its additional structure by Tb(S) and call it the Torelli building of S. The main result of this paper shows that Torelli buildings of surfaces of genus >5 have only obvious automorphisms, and identifies the group of automorphisms of Tb(S). Namely, we prove that for such surfaces S every automorphism of Tb(S) is induced by a diffeomorphism of S. This theorem is an analogue of the theorem of the second author [I7, I8] about authomorphisms of complexes of curves C(R). The proof is based on this theorem about automorphisms of complexes of curves, and uses methods inspired by its proof [I7, I8].