From symmetric spaces to buildings, curve complexes and outer spaces

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 ${\mathrm{\Delta }}_{\mathbb{Q}}\left(\mathbf{G}\right)$ of a semisimple linear algebraic group $\mathbf{G}$ defined over $\mathbb{Q}$, (2) a parametrization by the simplices of ${\mathrm{\Delta }}_{\mathbb{Q}}\left(\mathbf{G}\right)$ of the boundary components of the Borel-Serre partial compactification ${\overline{X}}^{BS}$ of the symmetric space $X$ associated with $\mathbf{G}$, which gives the Borel-Serre compactification of the quotient of $X$ by every arithmetic subgroup $\mathrm{\Gamma }$ of $\mathbf{G}\left(\mathbb{Q}\right)$, (3) and a realization of ${\overline{X}}^{BS}$ by a truncated submanifold $X_T$ of $X$. We then explain similar results for the curve complex $\mathcal{C}\left(S\right)$ of a surface $S$, Teichmüller spaces $T_g$, truncated submanifolds $T_g\left(\epsilon \right)$, and mapping class groups ${Mod}_g$ of surfaces. Finally, we recall the outer automorphism groups $Out\left(F_n\right)$ of free groups $F_n$ and the outer spaces $X_n$, construct truncated outer spaces $X_n\left(\epsilon \right)$, and introduce an infinite simplicial complex, called the core graph complex and denoted by $\mathcal{C}\mathcal{G}\left(F_n\right)$, and we then parametrize boundary components of the truncated outer space $X_n\left(\epsilon \right)$ by the simplices of the core graph complex $\mathcal{C}\mathcal{G}\left(F_n\right)$. 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.