Strongly transitive actions on Euclidean buildings

Abstract

We prove a decomposition result for a group $G$ acting strongly transitively on the Tits boundary of a Euclidean building. As an application we provide a local to global result for discrete Euclidean buildings, which generalizes results in the locally compact case by Caprace--Ciobotaru and Burger--Mozes. Let $X$ be a Euclidean building without cone factors. If a group $G$ of automorphisms of $X$ acts strongly transitively on the spherical building at infinity $\partial X$, then the $G$-stabilizer of every affine apartment in $X$ contains all reflections along thick walls. In particular $G$ acts strongly transitively on $X$ if $X$ is simplicial and thick.