### On Grothendieck's pairing of component groups in the semistable reduction case.

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We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation...

Inspired by Manin’s approach towards a geometric interpretation of Arakelov theory at infinity, we interpret in this paper non-Archimedean local intersection numbers of linear cycles in ${\mathbb{P}}^{n-1}$ with the combinatorial geometry of the Bruhat-Tits building associated to $PGL\left(n\right)$.

We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group $\mathrm{G}$ over a suitable non-Archimedean field $k$ we define a map from the Bruhat-Tits building $\mathcal{B}(\mathrm{G},k)$ to the Berkovich analytic space ${\mathrm{G}}^{\mathrm{an}}$ associated with $\mathrm{G}$. Composing this map with the projection of ${\mathrm{G}}^{\mathrm{an}}$ to its flag varieties, we define a family of compactifications of $\mathcal{B}(\mathrm{G},k)$. This generalizes results by Berkovich in the case of split groups. Moreover,...

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