Biaffine polar spaces (Q2844286)

Let \(\Pi\) be a thick polar space of rank \(n\geq 3\). Pick a hyperplane \(F\) of \(\Pi\) and \(H\) of \(\Pi^*\). Define the elements of a biaffine polar space \(\Gamma\) to be those elements of \(\Pi\) which are not contained in \(F\), or dually in \(H\).NEWLINENEWLINETheorem 4.8. Suppose a thick polar space \(\Pi\) is not \(W(2n-1,2)\cong Q(2n,2)\), \(W(2n-1,3)\), \(Q^{-}\)\((2n-1,2)\) or \(H(2n-1,2^2)\). If \(\Gamma\) has rank at least 4, then \(\Gamma\) is simply connected.NEWLINENEWLINETheorem 5.3. Suppose a thick polar space \(\Pi\) has rank three. Then the biaffine polar space \(\Gamma\) is simply connected, provided \(\Pi\) is not \(W(5,q)\), \(q\leq 4\), \(Q(6,q)\), \(q\leq 4\), \(Q^{-}\)\((7,2)\) or \(H(5,q^2)\), \(q\leq 4\).NEWLINENEWLINEA construction that leads to flag-transitive examples (Section 6) is given.