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Register a new userAn intrinsic characterization of Bruhat-Tits buildings inside analytic groups
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Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.
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Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.
In cite{FGLNP}, Fox, Gromov, Lafforgue, Naor and Pach, in a respond to a question of Gromov cite{G}, constructed bounded degree geometric expanders, namely, simplical complexes having the affine overlapping property. Their explicit constructions are finite quotients of $tilde{A_d}$-buildings, for $dgeq 2$, over local fields. In this paper, this result is extended to general high rank Bruhat-Tits buildings.
Let $(W,S)$ be a finite Weyl group and let $win W$. It is widely appreciated that the descent set D(w)={sin S | l(ws)<l(w)} determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset $W^J$ where $Jsubseteq S$. Our main results here include the identification of a certain subset $S^Jsubseteq W^J$ that convincingly plays the role of $Ssubseteq W$, at least from the point of view of descent sets and related geometry. The point here is to use this resulting {em descent system} $(W^J,S^J)$ to explicitly encode some of the geometry and combinatorics that is intrinsic to the poset $W^J$. In particular, we arrive at the notion of an {em augmented poset}, and we identify the {em combinatorially smooth} subsets $Jsubseteq S$ that have special geometric significance in terms of a certain corresponding torus embedding $X(J)$. The theory of $mathscr{J}$-irreducible monoids provides an essential tool in arriving at our main results.
We construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to Bruhat-Tits buildings, for which we also provide a Tannakian description.
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.
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