# The cone topology on masures

## TL;DR

This paper introduces a cone topology on the twin building at infinity of masures, establishing its properties and linking strong transitivity of group actions on masures to their actions on the twin building at infinity.

## Contribution

It defines and studies the cone topology on the twin building at infinity of masures and relates strong transitivity of group actions on masures to their actions on the twin building.

## Key findings

- The cone topology has favorable properties for topological twin buildings.
- Strong transitivity of a group on a masure is equivalent to strong transitivity on the twin building at infinity under certain conditions.
- Existence of strongly regular hyperbolic automorphisms with good dynamical properties.

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