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Triple factorisations of finite groups $G$ of the form $G=PQP$ are essentialin the study of Lie theory as well as in geometry. Geometrically, each triplefactorisation $G=PQP$ corresponds to a $G$-flag transitive point/line geometrysuch that `each pair of points is incident with at least one line'. We callsuch a geometry \emph{collinearly complete}, and duality (interchanging theroles of points and lines) gives rise to the notion of \emph{concurrentlycomplete} geometries. In this paper, we study triple factorisations of thegeneral linear group $\mathrm{GL}(V)$ as $PQP$ where the subgroups $P$ and $Q$either fix a subspace or fix a decomposition of $V$ as $V_1\oplus V_2$ with$\dim(V_{1})=\dim(V_{2})$.

Triple factorisations of the general linear group and their associated geometries