Some Polar Towers

Abstract

In this paper, we mainly consider four infinite towers of locally finite diagram geometries, and show that they are characterized by their diagram and the property of having a flag-transitive automorphism group. They form polar towers; that is, they are successive circle extensions of classical generalized quadrangles (series {Gdk;;>l of (ck. Cz)-geometries, Gk being a point residue of Gk + l ) , and possess flag-transitive automorphism groups. (The name goes back to a remark in [6]). Their residues of type (cl . Cz) can be viewed as locally polar spaces (see [6, Remarks (1.5)(3) and (4), p. 369]). We classify infinite towers of (c • Cz)-geometries with prescribed (c' . Cz)-residues, i = 1, resp. 2, resp. 3. The results of this paper, together with the classification of locally polar spaces [6,9,22] and results from [20] and [14], yield a classification of all flag-transitive (c • Cn)-geometries, including geometries related to the sporadic groups C03 , MeL, HS, Suz, COl and the Fischer groups, up to three cases of small rank, which will be dealt with in subsequent work. Here, a (c • Cz)-geometry is a residually connected geometry (of rank k + 2), with diagram