Volume 13, issue 3 (2013)

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A geometric construction of panel-regular lattices for buildings of types $\wtilde A_2$ and $\wtilde C_2$

Jan Essert

Algebraic & Geometric Topology 13 (2013) 1531–1578
Abstract

Using Singer polygons, we construct locally finite affine buildings of types ${Ã}_{2}$ and ${\stackrel{̃}{C}}_{2}$ 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 ${\stackrel{̃}{C}}_{2}$–buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices for buildings of type ${\stackrel{̃}{C}}_{2}$. Integral and rational group homology for the lattices is also calculated.

Keywords
affine buildings, lattices, exotic buildings, group theory, complexes of groups
Mathematical Subject Classification 2010
Primary: 20E42, 20F65, 22E40
Secondary: 20J06