Four-Dimensional Compact Projective Planes with a Solvable Collineation Group

Abstract

We consider a four-dimensional compact projective plane \(\pi = (\mathfrak{B},\mathfrak{L})\) whose collineation group Σ is six-dimensional and solvable with a nilradical N isomorphic to Nil×R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that Σ fixes a flag p ∈ W, acts transitively on \(\mathfrak{L}_p \backslash \{ W\}\) and fixes no point in the set W\p. Under these conditions, we will prove that either Σ contains a three-dimensional group of elations or Σ acts doubly transitively on \(\mathfrak{L}_p \backslash \{ W\}\).