A family of biaffine geometries and their resulting amalgams

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http://eprints.org/relation/isVolatileVersionOf http://etheses.bham.ac.uk/id/document/3733 http://eprints.org/relation/ispreviewThumbnailVersionOf http://etheses.bham.ac.uk/id/document/3733 http://eprints.org/relation/isVersionOf http://etheses.bham.ac.uk/id/document/3733 archive 4031 disk0/00/00/16/26 2011-06-23 10:21:48 2019-01-23 09:54:33 2011-07-19 16:39:18 thesis show 0 McInroy Justin Fergus Shpectorov Sergey University of Birmingham A family of biaffine geometries and their resulting amalgams unpub QA 10col_ephy 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program to calculate the fundamental group of a given geometry. 2011-07 2010-09 graduation eng author unchecked This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. public 171 University of Birmingham School of Mathematics d_ph FALSE epsrc 2011-07 graduation