Innovations in Incidence Geometry

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 ∆Q(G) of a semisimple linear algebraic group G defined over Q, (2) a parametrization by the simplices of∆Q(G) of the boundary components of the Borel-Serre partial compactification X BS of the symmetric space X associated with G, which gives the Borel-Serre compactification of the quotient of X by every arithmetic subgroup Γ of G(Q), (3) and a realization of X BS by a truncated submanifold XT of X. We then explain similar results for the curve complex C(S) of a surface S, Teichmüller spaces Tg, truncated submanifolds Tg(ε), and mapping class groups Modg of surfaces. Finally, we recall the outer automorphism groups Out(Fn) of free groups Fn and the outer spaces Xn, construct truncated outer spaces Xn(ε), and introduce an infinite simplicial complex, called the core graph complex and denoted by CG(Fn), and we then parametrize boundary components of the truncated outer space Xn(ε) by the simplices of the core graph complex CG(Fn). 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.