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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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