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A compactification of the Bruhat-Tits building. (English) Zbl 0935.20034
Lecture Notes in Mathematics. 1619. Berlin: Springer-Verlag. vii, 152 p. (1995).
The main object of the book under review is the Bruhat-Tits building $$X(G)$$ of a connected reductive group $$G$$ defined over a local field $$K$$. The author regards $$X(G)$$ as a $$p$$-adic analogue of a symmetric space, with an eye towards compactifying it. He constructs such a compactification $$\overline X(G)$$ being inspired by a polyhedral construction due to Borel-Serre. In this sense, the present construction generalizes that of P. Gérardin [in: Operator algebras and group representations, Vol. I, Monogr. Stud. Math. 17, 208-221 (1984; Zbl 0536.22018)]. The main idea is as follows: first, to compactify an apartment $$A$$ of $$X(G)$$ (this is done in §2 by accomodating the classical approach of A. Ash, D. Mumford, M. Rapoport, and Y. S. Tai [Smooth compactification of locally symmetric varieties (Math. Sci. Press, Brookline, MA) (1975; Zbl 0334.14007)] to the $$p$$-adic case), and then to use the obtained compactification $$\overline A$$ as a “local model” for compactifying $$X(G)$$. Namely, viewing $$X(G)$$ as the set of equivalence classes of $$G(K)\times A$$ modulo a suitable equivalence relation, one can define $$\overline X(G)$$ as the set of equivalence classes of $$G(K)\times\overline A$$ modulo a natural extension of this relation. This construction is described in detail in §14 (assuming that the residue field of $$K$$ is finite) and is illustrated in §15 by the example $$G=\text{SL}_n$$.
One should note that the exposition is almost self-contained. In particular, the author reproduces an essential part of the Bruhat-Tits theory. Among his technical innovations it is worth pointing out systematic use of the Néron-Raynaud integral models of algebraic tori (see Chapter 10 of S. Bosch, W. Lütkebohmert, and M. Raynaud [Néron models (Springer, Berlin) (1990; Zbl 0705.14001)]). Another interesting approach is used in the construction of an integral model of $$G$$: instead of the representation-theoretic method of Bruhat-Tits, the author employs birational group laws like in Chapter 5 of the above cited book on Néron models.

##### MSC:
 20G25 Linear algebraic groups over local fields and their integers 20E42 Groups with a $$BN$$-pair; buildings 20-02 Research exposition (monographs, survey articles) pertaining to group theory
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