Strongly transitive actions on Euclidean buildings

TL;DR

This paper establishes a decomposition for groups acting strongly transitively on Euclidean buildings' boundaries and derives local-to-global results, extending prior work to more general, possibly non-locally compact, settings.

Contribution

It introduces a new decomposition theorem for groups acting strongly transitively on Euclidean buildings' boundaries and generalizes local-to-global results beyond the locally compact case.

Findings

Group actions on Euclidean buildings can be decomposed into simpler components.

Strong transitivity on the boundary implies strong transitivity on the entire building under certain conditions.

The results extend known theorems to non-locally compact 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.