Strongly transitive actions on affine ordered hovels

Abstract

A hovel is a generalization of the Bruhat-Tits building that is associated to an almost split Kac-Moody group G over a non-Archimedean local field. In particular, G acts strongly transitively on its corresponding hovel $\Delta$ as well as on the building at infinity of $\Delta$, which is the twin building associated to G. In this paper we study strongly transitive actions of groups that act on affine ordered hovels $\Delta$ and give necessary and sufficient conditions such that the strong transitivity of the action on $\Delta$ is equivalent to the strong transitivity of the action of the group on its building at infinity $\partial \Delta$. Along the way a criterion for strong transitivity is given and the cone topology on the hovel is introduced. We also prove the existence of strongly regular hyperbolic automorphisms of the hovel, obtaining thus good dynamical properties on the building at infinity $\partial \Delta$.