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Bruhat-Tits Buildings and Analytic Geometry

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Berkovich Spaces and Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2119))

Abstract

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group.

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Notes

  1. 1.

    Though the notion is taken from [BrT72], the terminology we use here is not the exact translation of the French “donnée radicielle” as used in [loc. cit.]: this is because we have already used the terminology “root datum” in the combinatorial sense of [SGA3]. Accordingly, we use the notation of [SGA3] instead of that of [BrT72], e.g. a root system is denoted by the letter R instead of \(\Phi\).

  2. 2.

    This notion was introduced by Berkovich, who used the adjective peaked [Ber90, 5.2]. Its study was carried on by Poineau, who preferred the adjective universal [Poi13].

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Acknowledgements

We warmly thank the organizers of the summer school “Berkovich 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 Tobias Schmidt for pointing out that Lemma A.10 of [RTW10] needed to be corrected.

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Authors and Affiliations

  1. CMLS École Polytechnique Route de Saclay, Palaiseau Cedex, 91128, France

    Bertrand Rémy

  2. Université de Lyon 1 Institut Camille Jordan, 43 boulevard du 11 novembre, 1918, F-69622, Villeurbanne cedex, France

    Bertrand Rémy & Amaury Thuillier

  3. Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, D-60325, Frankfurt a. M., Germany

    Annette Werner

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  1. Bertrand Rémy
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Correspondence to Bertrand Rémy .

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Editors and Affiliations

  1. Pierre and Marie Curie University, Paris, France

    Antoine Ducros

  2. Ecole Polytechnique, Palaiseau Cedex, France

    Charles Favre

  3. University of Leuven, Heverlee, Belgium

    Johannes Nicaise

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Rémy, B., Thuillier, A., Werner, A. (2015). Bruhat-Tits Buildings and Analytic Geometry. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_5

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