$G$ is always cocompact in $\operatorname{Aut}(X)$ and it is hence conceivable that our lattice (or a finite-index subgroup) is already contained in $G$. Then Margulis’ Arithmeticity [@Mar:DSS:91] asserts that the lattice is *arithmetic*, that is, comes from an algebraic construction. Our construction would then provide simple presentations for arithmetic lattices, along with a very explicit description of the action on the building.

For the construction, we use *Singer polygons*, generalised polygons with a point-regular automorphism group, to construct small complexes of groups. The local developments of these complexes of groups are cones over the generalised polygons with which we have started. These cones are automatically non-positively curved in a natural metric, so the complexes of groups are developable by a theorem of Bridson and Haefliger in [@BH:NPC:99]. Then, by a recognition theorem by Charney and Lytchak in [@CL:MC:01], we know that their universal covers are buildings. The fundamental groups of these complexes of groups are then uniform lattices on two-dimensional locally finite affine buildings.

In principle, our construction might even extend to lattices in buildings of type $\tilde G_2$, but unfortunately there are no known Singer hexagons.

There are three geometric constructions of chamber-transitive lattices in buildings of type $\tilde 