A finiteness theorem in the Bruhat-Tits building: an application of Landvogt's embedding theorem

AbstractLet G be a connected reductive group over Qp and let B(G, Qp) be the extended Bruhat-Tits building of G over Qp. Let L be the completion of the maximal unramified extension of Qp and let B(G, L) be the building of G over L. By the theorem of Bruhat and Tits, B(G, Qp) may be identified with the fixed point set of the Frobenius automorphism σ acting on B(G, L). A special case of our main result states that for any c>0 there exists C > 0 with the property that any point x σ B(G, L) with distance d(x, σ(x)) < c is at distance < C from B(G, Qp). The results in this paper constitute a qualitative generalization of a result of Drinfeld