Abstract
In this article, we explain how spherical Tits buildings arise naturally and play a basic role in studying many questions about symmetric spaces and arithmetic groups, why Bruhat-Tits Euclidean buildings are needed for studying S-arithmetic groups, and how analogous simplicial complexes arise in other contexts and serve purposes similar to those of buildings.
We emphasize the close relationships between the following: (1) the spherical Tits building of a semisimple linear algebraic group defined over , (2) a parametrization by the simplices of of the boundary components of the Borel-Serre partial compactification of the symmetric space associated with , which gives the Borel-Serre compactification of the quotient of by every arithmetic subgroup of , (3) and a realization of by a truncated submanifold of . We then explain similar results for the curve complex of a surface , Teichmüller spaces , truncated submanifolds , and mapping class groups of surfaces. Finally, we recall the outer automorphism groups of free groups and the outer spaces , construct truncated outer spaces , and introduce an infinite simplicial complex, called the core graph complex and denoted by , and we then parametrize boundary components of the truncated outer space by the simplices of the core graph complex . This latter result suggests that the core graph complex is a proper analogue of the spherical Tits building.
The ubiquity of such relationships between simplicial complexes and structures at infinity of natural spaces sheds a different kind of light on the importance of Tits buildings.
Citation
Lizhen Ji. "From symmetric spaces to buildings, curve complexes and outer spaces." Innov. Incidence Geom. 10 33 - 80, 2009. https://doi.org/10.2140/iig.2009.10.33
Information