51 research outputs found

    Affine twin R-buildings

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    AbstractIn this paper, we define twinnings for affine R-buildings. We thus extend the theory of simplicial twin buildings of affine type to the non-simplicial case. We show how classical results can be extended to the non-discrete case, and, as an application, we prove that the buildings at infinity of a Moufang twin R-building have the induced structure of a Moufang building. The latter is not true for ordinary “Moufang” R-buildings

    Open subgroups of the automorphism group of a right-angled building

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    We study the group of type-preserving automorphisms of a right-angled building, in particular when the building is locally finite. Our aim is to characterize the proper open subgroups as the finite index closed subgroups of the stabilizers of proper residues. One of the main tools is the new notion of firm elements in a right-angled Coxeter group, which are those elements for which the final letter in each reduced representation is the same. We also introduce the related notions of firmness for arbitrary elements of such a Coxeter group and nn-flexibility of chambers in a right-angled building. These notions and their properties are used to determine the set of chambers fixed by the fixator of a ball. Our main result is obtained by combining these facts with ideas by Pierre-Emmanuel Caprace and Timoth\'ee Marquis in the context of Kac-Moody groups over finite fields, where we had to replace the notion of root groups by a new notion of root wing groups.Comment: 29 page

    Flat rank of automorphism groups of buildings

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    The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities: alg-rk(W)\le flat-rk(G)\le rk(|W|\_0). Here, alg-rk(W) is the maximal Z\mathbb{Z}-rank of abelian subgroups of W, and rk(|W|\_0) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|\_0. We can prove these inequalities under weaker assumptions. We also show that for any integer n \geq 1 there is a topologically simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G)=n and which is not linear

    Word maps in Kac-Moody setting

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    The paper is a short survey of recent developments in the area of word maps evaluated on groups and algebras. It is aimed to pose questions relevant to Kac--Moody theory.Comment: 16 pag

    Contraction groups in complete Kac-Moody groups

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    Let G be an abstract Kac-Moody group over a finite field and G the closure of the image of G in the automorphism group of its positive building. We show that if the Dynkin diagram associated to G is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in G which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.
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