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

Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $\mathrm{\Delta }$ of dimension $\ge 2$ that acts strongly transitively on $\mathrm{\Delta }$ is Moufang. If moreover $\mathrm{\Delta }$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\text{G}(k{)}^{+}$ of the $k$-rational points $\text{G}(k)$ of the isotropic simple algebraic group $\text{G}$ over a non-Archimedean local field $k$ associated with $\mathrm{\Delta }$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $\mathrm{\Delta }$, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of $\text{Aut}(\mathrm{\Delta })$ 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 $K{A}^{+}K$, where $K$ is a maximal compact subgroup, and the important fact that $A^{+}$ is an abelian maximal sub-semi-group.