A family of biaffine geometries and their resulting amalgams

McInroy, Justin Fergus (2011). A family of biaffine geometries and their resulting amalgams. University of Birmingham. Ph.D.

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Abstract

Let $\Pi$ be a thick polar space of rank $n$ at least three. Pick a hyperplane $F$ of $\Pi$ and $H$ of $\Pi$ $^{\ast}$ . Define the elements of a biaffine polar space $\Gamma$ to be those elements of $\Pi$ which are not contained in $F$ , or dually in $H$ . We show that $\Gamma$ is a non empty geometry which is simply connected, except for a few small exceptions for $\Pi$ . We give two pairs of examples with ag-transitive groups, which lead to amalgam results for recognising either one of $q$ $^6$ : $SU$ $_3$ $(q)$ or $G$ $_2$ $(q)$ , or one of $q$ $^7$ : $G$ $_2$ $(q)$ or $Spin$ $^7$ $(q)$ . Also, we give details of a computer program to calculate the fundamental group of a given geometry.