A criterion concerning Singer groups of generalized quadrangles, and construction of uniform lattices in $\widetilde{\mathbf{C}_2}$-buildings

Abstract

We describe a simple criterion to construct Singer groups of Payne-derived generalized quadrangles, yielding, as a corollary, a classification of Singer groups of the classical Payne-derived quadrangles in any characteristic. This generalizes recent constructions of Singer groups of these quadrangles that were presented in a paper by Bamberg and Giudici. In the linear case, and several other cases, our classification is complete. Contrary to what seemed to be a common belief, we show that for the classical Payne-derived quadrangles, the number of different Singer groups is extremely large, and even bounded below by an exponential function of the order of the ground field. Our results have direct applications to the theory of $\widetilde{\mathbf{C}_2}$-buildings, which are explained at the end of the paper.