The cone topology on masures

Corina Ciobotaru (1) , Bernhard Mühlherr (2) , Guy Rousseau (3) , Auguste Hébert (4)

1 UNIFR - Université de Fribourg = University of Fribourg
2 JLU - Justus-Liebig-Universität Gießen = Justus Liebig University
3 IECL - Institut Élie Cartan de Lorraine
4 AGL - Algèbre, géométrie, logique

Masures are generalizations of Bruhat–Tits buildings and the main examples are associated with almost split Kac–Moody groups G over non-Archimedean local fields. In this case, G acts strongly transitively on its corresponding masure ∆ as well as on the building at infinity of ∆, which is the twin building associated with G. The aim of this article is twofold: firstly, to introduce and study the cone topology on the twin building at infinity of a masure. It turns out that this topology has various favorable properties that are required in the literature as axioms for a topological twin building. Secondly, by making use of the cone topology, we study strongly transitive actions of a group G on a masure ∆. Under some hypotheses, with respect to the masure and the group action of G, we prove that G acts strongly transitively on ∆ if and only if it acts strongly transitively on the twin building at infinity ∂∆. Along the way a criterion for strong transitivity is given and the existence and good dynamical properties of strongly regular hyperbolic automorphisms of the masure are proven.

Mots clés: masure; cone topology; building

Domaines: Théorie des groupes [math.GR]

Corina Ciobotaru, Bernhard Mühlherr, Guy Rousseau, Auguste Hébert. The cone topology on masures. Advances in Geometry, 2020, 20, pp.1-28. ⟨10.1515/advgeom-2019-0020⟩. ⟨hal-01479708v2⟩