#### Vol. 10, No. 1, 2009

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From symmetric spaces to buildings, curve complexes and outer spaces

### Lizhen Ji

Vol. 10 (2009), No. 1, 33–80
##### 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 ${\Delta }_{ℚ}\left(\mathbf{G}\right)$ of a semisimple linear algebraic group $\mathbf{G}$ defined over $ℚ$, (2) a parametrization by the simplices of ${\Delta }_{ℚ}\left(\mathbf{G}\right)$ 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 $\Gamma$ of $\mathbf{G}\left(ℚ\right)$, (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 $\mathsc{C}\left(S\right)$ 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 $\mathsc{C}\mathsc{G}\left({F}_{n}\right)$, and we then parametrize boundary components of the truncated outer space ${X}_{n}\left(\epsilon \right)$ by the simplices of the core graph complex $\mathsc{C}\mathsc{G}\left({F}_{n}\right)$. 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.

##### Mathematical Subject Classification 2000
Primary: 14L30, 20E06, 20E42, 32G15