Torelli buildings and their automorphisms

TL;DR

This paper introduces Torelli buildings, a new combinatorial structure associated with surfaces, and proves that their automorphisms are induced by surface diffeomorphisms for genus at least 5.

Contribution

It defines Torelli buildings for surfaces and establishes that their automorphisms correspond exactly to surface diffeomorphisms, revealing their symmetry structure.

Findings

Torelli buildings are simplicial complexes with additional structure.

Automorphisms of Torelli buildings are induced by surface diffeomorphisms for genus ≥ 5.

The automorphism group of the Torelli building is identified with the surface's mapping class group.

Abstract

In this paper we introduce, for each closed orientable surface, an analogue of Tits buildings adjusted to investigation of the Torelli group of this surface. It is a simplicial complex with some additional structure. We call this complex with its additional structure the Torelli building of the surface in question. The main result of this paper shows that Torelli buildings of surfaces of genus at least 5 have only obvious automorphisms, and identifies its group of automorphisms. Namely, we prove that for such a surface every automorphism of its Torelli building is induced by a diffeomorphism of the surface. This theorem about automorphisms of Torelli buildings is intended for applications to automorphisms and virtual automorphisms of Torelli groups. The latter results will be presented on some other occasion. All these results were announced in arXiv:math/0311123.