# Triality, two geometries and one amalgam non-uniqueness result

A polar space $\Pi$ is a geometry whose elements are the totally isotropic subspaces of a vector space $V$ with respect to either an alternating, Hermitian, or quadratic form. We may form a new geometry $\Gamma$ by removing all elements contained in either a hyperplane $F$ of $\Pi$, or a hyperplane $H$ of the dual $\Pi*$. This is a \emph{biaffine polar space}.

We will discuss two specific examples arising from the triality in $O+_8(q)$. By considering the stabilisers of a maximal flag, we get an amalgam, or “glueing”, of groups for each example. However, the two examples have “similar” amalgams – this leads to a group recognition result for their automorphism groups, $q^7:G_2(q)$ and $Spin_7(q)$.

This talk is part of the Junior Algebra/Logic/Number Theory seminar series.