# Imaginary geometries

## Keywords:

Long root geometry, root groups, spherical buildings, Moufang buildings## Abstract

In this paper, we axiomatize the geometries obtained from the long root subgroup geometries by taking as new lines the so-called imaginary lines. A generic such line is the union of the orbits of the centers of the two root groups corresponding to two opposite long roots, which share at least two points. This extends characterizations of Cuypers and Hall on copolar spaces, who treated the quadrangular case. Here, we treat the remaining case, the hexagonal one. Our results hold over any field of size at least 5 and characteristic different from 2.

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*Modern Mathematical Methods*,

*1*(1), 43–92. Retrieved from https://modernmathmeth.com/index.php/pub/article/view/11

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