A geometric construction of panel-regular lattices for buildings of types $\tilde{A}_2$ and $\tilde{C}_2$

Abstract

Using Singer polygons, we construct locally finite affine buildings of types ${Ã}_2$ and ${\tilde{C}}_2$ that admit uniform lattices acting regularly on panels. For type ${Ã}_2$, these cover all possible buildings admitting panel-regular lattices. All but one of the ${\tilde{C}}_2$–buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices for buildings of type ${\tilde{C}}_2$. Integral and rational group homology for the lattices is also calculated.