Compact three-dimensional elation quadrangles (Q1591220)

A generalized quadrangle \(Q\) is an elation quadrangle with elation group \(E\leq \Aut Q\) if there is a point \(c\) such that \(E\) fixes all lines passing through \(c\) and acts sharply transitively on the points not collinear with \(c\). The author determines all compact connected 3-dimensional elation generalized quadrangles. Either \(E\) is the nilpotent 3-dimensional (Heisenberg) group \(H\) and \(Q\) is the real symplectic quadrangle, or \(E= \mathbb{R}^3\) and \(Q\) is a translation generalized quadrangle of Tits type (constructed from an oval in the real projective plane) [compare \textit{M. Joswig}, Result. Math. 38, 72-87 (2000; Zbl 0956.51005)]. The proof starts from the observation that the elation group is a 3-dimensional Lie group and that the stabilizers of all points \(p\neq c\) on a given line through \(c\) form a conjugacy class. The author then proceeds to determine all 3-dimensional Lie groups having infinitely many conjugacy classes of 2-dimensional subgroups; apart from \(H\) and \(\mathbb{R}^3\), only the dilatation group \(\text{Dil}_2\mathbb{R}\) (extension of \(\mathbb{R}^2\) by the positive homothetics) and \(\mathbb{R}\times \text{Dil}_1 \mathbb{R}\) have this property. To exclude the last group turns out to be particularly difficult; searching for a contradiction, one is led to construct a family of subgroups that lacks only one of the defining properties of a 4-gonal family. Wherever possible, the author proves his results without making assumptions on the topological dimension of the generalized quadrangle, sometimes even without assuming a topology at all.