On \(C_2. L\)-geometries (Q1392730)

Let \(\Gamma\) be a \(C_2.L\)-geometry (the elements are points, lines and quads with respective residues linear spaces, generalized digons and generalized quadrangles), not of type \(C_3\). The authors have found in the literature (putting together results from various sources) a series of sufficient conditions for \(\Gamma\) to be the truncation of a higher dimensional geometry \(\overline\Gamma\) (of type \(C_2.A_2.L)\) and for the residue of each point of \(\Gamma\) to be the point-line space arising from a projective space; if the latter is finite dimensional, then \(\overline \Gamma\) itself is a (quotient of \(a)\) \(C_n\)-building. These sufficient conditions are: (1) all lines are thick, (2) given a line \(L\) and a point \(p\) at distance \(d\leq 3\) from each other in the collinearity graph of \(\Gamma\) (where we view \(L\) as the set of points incident with \(L)\), then there is a unique point on \(L\) at distance \(d\) from \(p\), and (3) given a quad \(\alpha\) and a point \(p\) at distance \(d\leq 2\) from each other in the collinearity graph of \(\Gamma\) (where we view \(\alpha\) as the set of points incident with \(\alpha)\), then there is a unique point on \(\alpha\) at distance \(d\) from \(p\). In the paper under review, the authors obtain the same conclusion by removing condition (1), and by only requiring (2) for \(d\leq 2\). They also look at some special cases (the finite case, the case of type \(C_2.c)\) and they also describe what happens is one deletes (2), but requires that all lines are thin. As an application, a similar theorem for \(L.L^*\)-geometries is proved (where now quotients of truncations of \(D_n\)-buildings are obtained). The proofs in the paper are purely geometric and combinatorial.