spaces” held in Paris in July 2010. We are grateful to the referee for many comments, corrections and some relevant questions, one of which led to Proposition 5.11. Finally, we thank Christophe Cornut for an interesting (electronic) discussion which prompted us to formulate and prove Theorem 5.7.

[**Conventions.**]{}  In this paper, as in [@Ber1], valued fields are assumed to be non-Archimedean and complete, the valuation ring of such a field $k$ is denoted by $k^\circ$, its maximal ideal is by $k^{\circ\circ}$ and its residue field by $\widetilde{k} =
k^\circ/k^{\circ \circ}$. Moreover a *local field* is a non-trivially valued non-Archimedean field which is locally compact for the topology given by the valuation (i.e., it is complete, the valuation is discrete and the residue field is finite).

Buildings and special linear groups {#s - SL(n) Bruhat-Tits}
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We first provide a (very quick) general treatment of Euclidean buildings; general references for this notion are [@RousseauGrenoble] and [@WeissAffine]. It is important for us to deal with the simplicial as well as the non-simplicial version of the notion of a Euclidean building because compactifying Bruhat-Tits buildings via Berkovich techniques uses huge valued fields. The second part illustrates these definitions for special linear groups; in 