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 BruhatTits Euclidean buildings are needed for studying Sarithmetic 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
${\Delta}_{\mathbb{Q}}\left(\mathbf{G}\right)$ of a semisimple linear
algebraic group
$\mathbf{G}$ defined
over
$\mathbb{Q}$, (2) a parametrization
by the simplices of
${\Delta}_{\mathbb{Q}}\left(\mathbf{G}\right)$
of the boundary components of the BorelSerre partial compactification
${\overline{X}}^{BS}$ of the symmetric
space
$X$
associated with
$\mathbf{G}$,
which gives the BorelSerre compactification of the quotient of
$X$ by every arithmetic
subgroup
$\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.
