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Some polar towers. (English) Zbl 0753.05016
Geometries related to 3-transposition groups are studied. More specifically, let \(n\geq 4\), and consider the graph \(\Gamma_ n\) whose vertices are the non-isotropic points of a non-degenerate Hermitian form on \(GF(4)^ n\) and in which adjacency is perpendicularity. Then the induced graph on the neighbours of a vertex is isomorphic to \(\Gamma_{n-1}\) (provided \(n\geq 5)\) and is the collinearity graph of the \(Sp(4,3)\) generalized quadrangle. Taking as objects the \(j\)-cliques of \(\Gamma_ n\) for \(j=1,2,\ldots,n-2\), \(n-3,n\), and symmetrised inclusion as incidence, one obtains a residually connected (r.c.) flag-transitive (f.t.) geometry of type \(c^{n-4}\cdot C_ 2\) (that is, a linear diagram of length \(n-2\) all of whose edges represent extensions, type \(c\), except for the last one, which represents a generalised quadrangle, type \(C_ 2)\).
Similar constructions with orthogonal forms on \(GF(3)^ n\) (where vertices are norm 1 points; two types of geometries, parametrised by the discriminant \(\varepsilon\) of the form) and with 2-sets from \(\{1,2,\ldots,2n\}\) also lead to r.c. f.t. geometries of the indicated type.
The author characterises these geometries as r.c. f.t. geometries with prescribed isomorphism types of the \(c^ i\cdot C_ 2\) residues, where \(i=1\) in the last case, \(i=1,3\) in the orthogonal cases with \(\varepsilon=1,-1\), respectively, and \(i=2\) in the Hermitian case \(\Gamma_ n\). The difficulties in low rank cases are related to the existence of proper covers.
An analogous result concerns the characterisation of a r.c. f.t. geometry of type \(c^ i\cdot C_ 3(i\geq 1)\) whose \(C_ 3\) residues (rank 3 polar geometries) are \(U(6,2^ 2)\) buildings. It leads to a known geometry connected with a Fischer group \(Fi_{21+i}\) for \(i=1,2,3\), or to a triple cover of the latter.
The paper ends with a construction similar to the one for \(\Gamma_ n\), but now starting from a Hermitian form on \(GF(9)^ n\), leading to a r.c. f.t. geometry of type \(c^{n-3}\cdot G_ 2\) (an extended generalised hexagon).

MSC:
05B25 Combinatorial aspects of finite geometries
51D20 Combinatorial geometries and geometric closure systems
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