The extremal symmetry of arithmetic simplicial complexes
نویسندگان
چکیده
Let K be a nonarchimedean local field, for example the p-adic numbers Qp (char(K) = 0) or the field of Laurent series over a finite field Fp((t)) (char(p) > 0) . Let G = PGLn(K), or more generally the K-points of any absolutely simple, connected, algebraic K-group of adjoint form. There is a natural way to associate to each cocompact lattice Γ in G a finite simplicial complex BΓ, as follows. Bruhat-Tits theory (see below) provides a contractible, rankK Gdimensional simplicial complex XG on which G acts by simplicial automorphisms. The lattice Γ acts properly discontinuously on XG with quotient a simplicial complex BΓ. 1 Margulis proved (see, e.g., [Ma]) that rankK G ≥ 2 implies that every lattice Γ in G is arithmetic. We also note that char(K) = 0 implies every lattice in G(K) is cocompact. In this paper we explore one aspect of the theme that, since the complex BΓ is constructed using number theory, it should have remarkable properties. Here we concentrate on the extremal nature of the symmetry of BΓ and all of its covers. Our first theorem shows that the simplicial structure of BΓ realizes all simplicial symmetries of any simplicial complex homeomorphic to BΓ. For any simplicial complex C we denote by Aut(C) the group of simplicial automorphisms of C. We denote by |C| the simplicial complex C thought of as a topological space, without remembering the simplicial structure.
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