Natural representations of some tilde and Petersen type geometries (Q1900004)

Given an abstract group geometry, the question naturally arises as to whether this geometry can be realized inside a classical projective geometry (or vector space). The geometries considered in this paper have three points on a line, so the underlying field of such a vector space must be \(\text{GF} (2)\). It is shown that for the tilde geometry of rank \(n\) there is a universal such representation, whose faithful part has dimension \(2^n (2^n -1)\). On the other hand, the Petersen geometry of rank 4 for the group \(3^{23} \cdot Co_2\) has no such representation. For a survey of other related results and applications, see the paper by \textit{A. A. Ivanov} and the author [Bull. Am. Math. Soc., New Ser. 31, No. 2, 173-184 (1994; Zbl 0817.20014)].