On the Petrials of thin rank 3 geometries (Q5954163)

In [Math. Comput. 68, 1631-1647 (1999; Zbl 0941.51010)] \textit{D. Leemans} showed that any thin regular geometry of rank three over a linear diagram can be seen as a regular and reflexive map and vice versa. Generalizing the definition of a Petrie polygon the authors extend the concept of a Petrial to thin regular rank three geometries. They show that the Petrial \(\Pi(\Gamma)\) of a thin regular residually connected rank three geometry \(\Gamma\) is residually connected, that \(\Pi(\Pi(\Gamma))=\Gamma\) and that \(\Gamma\) and \(\Pi(\Gamma)\) have the same automorphism group. This is used to show that if \(\Pi(\Gamma)\) also is a thin regular geometry, then \(\Gamma\) has a linear diagram. The converse is not true however and counter-examples may be found in Leemans' article [loc. cit.].