Non-Archimedean analytic spaces (Q354124)

The paper under review offers a short introduction to non-Archimedean analytic spaces in the sense of [\textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (AMS) (1990; Zbl 0715.14013)] and their applications to Bruhat-Tits buildings. After recalling the definition of a non-Archimedean valued field, the author gives a detailed description of the Berkovich unit disc before saying a word about the projective line and more general spaces. The most original part follows: an exposition of the relation between Berkovich spaces and Bruhat-Tits buildings at a rather elementary level, first on the example of \(\mathrm{SL}_{2}\) and then more generally (see Berkovich, [loc. cit.], chapter 5 and \textit{B. Rémy, A. Thuillier} and \textit{A. Werner} [Ann. Sci. Éc. Norm. Supér. (4) 43, No. 3, 461--554 (2010; Zbl 1198.51006)] for a complete treatment). Starting from the building associated to a semisimple algebraic group \(G\) over a non-Archimedean field \(K\), the author manages to give an overview of the construction of its embedding into the Berkovich space associated to \(G\) and to explain how it may be used in order to compactify it.