Parameters of translation generalized quadrangles

Abstract

In this note we show that the algebraic parameters of a linear translation generalized quadrangle are not restricted. This is done with a free construction of fourgonal families on vector spaces. Secondly we prove that a compact translation generalized quadrangle can only have the topological parameters $(1,t)$, $(2,2)$, $(3,4t)$ or $(7,8t)$ for $t\in\mathbb N$. This is achieved by determining the possible dimensions of the elements of continuous partial spreads which satisfy a certain planarity condition.