over a non-Archimedean local field.

The special case of Corollary \[corollary:affine\] where $X$ is of type $\tilde{A}_2$ was obtained by Van Maldeghem and Van Steen in [@VM]\*[Main Theorem]{}, while the general case has been obtained by Caprace and Monod in [@CapraceMonod]\*[Corollary E]{} using CAT($0$) geometry. Our approach to Theorem \[theorem:Characterization\] is different, but was partly inspired by theirs: instead of the CAT($0$) Levi decomposition used in their work, we use a purely algebraic counterpart of that result, due to Baumgartner and Willis [@Baumgartner], and valid in the realm of totally disconnected locally compact groups.

The basic result that makes the structure theory of locally compact groups available in the study of compact buildings is the theorem, due to Burns–Spatzier and valid for all thick irreducible compact spherical building $\Delta$ of rank $\geq 2$, asserting that the topological automorphism group $\operatorname{\mathrm{Auttop}}(\Delta)$ is a locally compact group, see [@Burns]\*[Theorem 2.1]{}. Equivalently, there exists an identity neighbourhood in the group $\operatorname{\mathrm{Auttop}}(\Delta)$ endowed with the compact-open topology that is compact. Our second main result is a compactness criterion for more general subsets of $\operatorname{\mathrm{Auttop}}(\Delta)$ in the rank $2$ case. Notice that we do not make any transitivity assumption on the automorphism group.

\[theorem:Criterion\] Let 