Strong transitivity, Moufang's condition and the Howe--Moore property

TL;DR

This paper proves that certain automorphism groups of affine buildings are Moufang and have the Howe--Moore property, revealing deep structural properties and classifications of these groups in geometric and algebraic contexts.

Contribution

It establishes Moufang properties for automorphism groups of affine buildings and generalizes the Howe--Moore property to a broad class of these groups, using novel proof techniques.

Findings

Automorphism groups of affine buildings are Moufang under certain conditions.

Topologically simple, strongly transitive groups have the Howe--Moore property.

Classification of certain automorphism groups as algebraic groups over local fields.

Abstract

Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $\Delta$ of dimension $\geq 2$ that acts strongly transitively on $\Delta$ is Moufang. If moreover $\Delta$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\G(k)^+$ of the $k$-rational points $\G(k)$ of the isotropic simple algebraic group $\G$ over a non-Archimedean local field $k$ associated with $\Delta$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $\Delta$, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of $\Aut(\Delta)$ has the Howe--Moore property. This proof is different than the strategy used so far in the literature and does not relay on the polar decomposition $KA^+K$, where $K$ is a maximal compact subgroup, and the important fact that $A^+$ is an abelian maximal sub-semi-group.