Peter Abramenko

Présentations de certaines BN-paires jumelées comme sommes amalgamées

Resume On donne une presentation de certaines BN -paires jumelees comme sommes amalgamees. Notre resultat generalise un theoreme de Tits sur les BN -paires spheriques. En utilisant la theorie des immeubles jumeles, les auteurs ont obtenu le resultat chacun par des methodes differentes.

Automorphisms of non-spherical buildings have unbounded displacement

If f is a nontrivial automorphism of a thick building Delta of purely infinite type, we prove that there is no bound on the distance that f moves a chamber. This has the following group-theoretic consequence: If G is a group of automorphisms of Delta with bounded quotient, then the center of G is trivial.

1-twinnings of buildings

Abstract. In this paper, we characterize twinnings of buildings by 1-twinnings and one further condition concerning twin apartments. Specialized to the spherical case, we obtain new characterizations of the opposition relation in such buildings. We also give a new description of the standard twin building of type

Transitivity properties for group actions on buildings

\widetilde{A}_{n-1}

Maps between buildings that preserve a given Weyl distance

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Walls in Coxeter complexes

Abstract We study two transitivity properties for group actions on buildings, called Weyl transitivity and strong transitivity. Following hints by Tits, we give examples involving anisotropic algebraic groups to show that strong transitivity is strictly stronger than Weyl transitivity. A surprising feature of the examples is that strong transitivity holds more often than expected.

Characterisations of generalized polygons and opposition in rank 2 twin buildings

Abstract Let Δ and Δ′ be two buildings of the same type (W, S), viewed as sets of chambers endowed with“distance” functions δ and δ′, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element ω e W, we study surjective maps ϕ : Δ → Δ′ with the property that δ(C, D) = ω if and only if Δ′ (ϕ(C), ϕ(D)) = ω. The result is that the restrictions of ϕ to all residues of certain spherical types—determined by ω—are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ϕ is an isomorphism. In particular, ϕ is an isomorphism if Δ and Δ′ are 2-spherical and every reduced expression of ω involves all elements of S.

Connectedness of Opposite-flag Geometries in Moufang Polygons

The starting point of this paper was the following question: Which walls in Coxeter complexes are Coxeter complexes in their own right? A complete answer to this question is given in the case of finite Coxeter complexes. In general, a sufficient criterion (depending on the entries of the Coxeter matrix) is derived which implies that walls of a certain type are always Coxeter complexes. It is studied how their Weyl groups are related to those of the original Coxeter complexes. Additionally, some statements being true for all walls are proved more generally for convex subcomplexes of Coxeter complexes.

Intersections of apartments

Abstract. In this paper, we characterize the natural opposition relation on the set of flags of a generalized polygon. We also investigate when a certain relation on any rank 2 geometry of finite diameter is equivalent to the opposition relation in a generalized polygon. As a consequence we obtain a new definition of generalized polygons. Finally, we also characterize the opposition relation in twin trees, which are the analogues of polygons with infinite diameter.

Buildings: Theory and Applications

We show that the geometry of the elements opposite a certain flag in a Moufang polygon is always connected, up to some small cases. This completes the determination of all Moufang polygons for which this geometry is disconnected.