Noncompact groups acting on topological translation planes
by
Harald Löwe
Technische Universität Braunschweig

We use techniques from Lie theory in order to study noncompact subgroups G of the reduced stabilizer of a 2l-dimensional translation plane (cp. H. Hähl's talk). The aim is to show that the possibilities for such groups are not far away from the classical (i.e. the real, complex, quaternion, and octonion) case. In particular, we present the following results:

If the Levi complements of G are not compact, then G is an almost direct product of Spin(m+2, 1), 1 <= m <= l, and a compact group. If m strictly exceeds l/2, then the plane is classical. For m=l/2, all possible planes have been classified (Betten, Hähl, Löwe).

Next, suppose that the radical R of G is not compact (notice that then the Levi complements are necessarily compact). Then G fixes a point s of the line W at infinity. Moreover, there exists a point w in W\{s} such that the stabilizer R_w contains a maximal torus T of R. According to Hähl's theorem on compression subgroups, we infer that R_w either equals T or is a direct product of T and a noncompact one-dimensional subgroup. Moreover, R is a semidirect product of R_w and a simply connected normal subgroup acting freely on W\{s}.

Date received: February 22, 2001