A family of biaffine geometries and their resulting amalgams

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.

Name dc:type.qualificationname

d_ph

Level dc:type.qualificationlevel

d_ph

Grantor dc:publisher.institution

University of Birmingham

Year dc:date.issued

2011

Author dc:creator

• McInroy, Justin Fergus

• QA Mathematics