Torelli buildings and their automorphisms

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.

[1]  Nathalie Wahl.  Homological stability for mapping class groups of surfaces , 2010, 1006.4476.

[2]  Nathalie Wahl.  Homological stability for the mapping class groups of non-orientable surfaces , 2006, math/0601310.

[3]  Mustafa Korkmaz,et al.  Automorphisms of complexes of curves on punctured spheres and on punctured tori , 1999 .

[4]  Feng Luo,et al.  Automorphisms of the complex of curves , 1999, math/9904020.

[5]  R. Hain.  Infinitesimal presentations of the Torelli groups , 1995, alg-geom/9512001.

[6]  R. Hain.  Torelli Groups and Geometry of Moduli Spaces of Curves , 1994, alg-geom/9403015.

[7]  G. Mess.  The Torelli groups for genus 2 and 3 surfaces , 1992 .

[8]  N. V. Ivanov.  Teichmüller modular groups and arithmetic groups , 1990 .

[9]  N. V. Ivanov.  Complexes of curves and the Teichmüller modular group , 1987 .

[10]  J. McCarthy.  Automorphisms of surface mapping class groups. A recent theorem of N. Ivanov , 1986 .

[11]  J. Harer.  The virtual cohomological dimension of the mapping class group of an orientable surface , 1986 .

[12]  John Harer,et al.  Stability of the homology of the mapping class groups of orientable surfaces , 1985 .

[13]  Dennis Johnson,et al.  The structure of the Torelli group I : A finite set of generators for J , 1983 .

[14]  Tara E. Brendle,et al.  Mapping class groups , 2017 .

[15]  Nikolai V. Ivanov,et al.  Automorphism of complexes of curves and of Teichmuller spaces : Interna , 1997 .

[16]  N. V. Ivanov.  Automorphisms of Teichmüller modular groups , 1988 .

[17]  N. V. Ivanov.  On the virtual cohomology dimension of the Teichmüller modular group , 1984 .

[18]  D. L. Johnson.  A survey of the Torelli group , 1983 .