A homogeneous $\tilde{A}_2$-building with a non-discrete automorphism group is Bruhat-Tits

Abstract

Let $\Delta$ be a locally finite thick building of type $\tilde{A}_2$. We show that, if the type-preserving automorphism group $\mathrm{Aut}(\Delta)^+$ of $\Delta$ is transitive on panels of each type, then either $\Delta$ is Bruhat--Tits or $\mathrm{Aut}(\Delta)$ is discrete. For $\tilde{A}_2$-buildings which are not panel-transitive but only vertex-transitive, we give additional conditions under which the same conclusion holds. We also find a local condition under which an $\tilde{A}_2$-building is ensured to be exotic (i.e.\ not Bruhat--Tits). It can be used to show that the number of exotic $\tilde{A}_2$-buildings with thickness $q+1$ and admitting a panel-regular lattice grows super-exponentially with $q$ (ranging over prime powers). All those exotic $\tilde{A}_2$-buildings have a discrete automorphism group.