The cone topology on masures

Abstract

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 $Δ$ as well as on the building at infinity of $Δ$, 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 $Δ$. Under some hypotheses, with respect to the masure and the group action of G, we prove that G acts strongly transitively on $Δ$ if and only if it acts strongly transitively on the twin building at infinity $\partial$$Δ$. 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.