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# A family of biaffine geometries and their resulting amalgams

McInroy, Justin Fergus (2011)
Ph.D. thesis, University of Birmingham.

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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.

Type of Work: Ph.D. thesis. Shpectorov, Sergey (Prof.) Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Mathematics QA Mathematics University of Birmingham 1626
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