A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

TL;DR

This paper introduces a geometric method to construct and describe lattices in affine buildings of types ~A_2 and ~C_2, providing explicit descriptions and new examples, especially for the exotic ~C_2 cases.

Contribution

It presents the first known lattice presentations in buildings of type ~C_2 using Singer polygons, with explicit constructions and homology calculations.

Findings

Constructed new affine buildings of types ~A_2 and ~C_2

Provided explicit lattice presentations, including for exotic ~C_2 buildings

Calculated integral and rational homology of the lattices

Abstract

Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.