On the Moufang condition for polar spaces

Abstract We provide direct constructions of nontrivial root elations of thick generalized quadrangles embedded in thick polar spaces of higher finite rank and show that all of these can be expressed as self-projectivities of length 4. This reveals a connection between the little projective groups and the special projectivity groups of thick polar spaces of finite rank explicitly. We also prove that the constructed root elations extend to the ambient polar space. Furthermore, we introduce point elations and line elations and use them to give an equivalent definition of the Moufang property for thick polar spaces, without using the notion of apartments. This, in turn, allows us to define the Moufang property for thick polar spaces of infinite rank. Consequently, we show that all thick polar spaces of finite and infinite rank are Moufang.