51 research outputs found
Affine twin R-buildings
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
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 -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
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 -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
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
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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