Some Infinite Families of Locally Partial Geometries Generalizing the Classical Laguerre Planes

Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β ⩾ 1, we construct inductively a series of rank n residually connected geometries Γ(n, β, Ω), n ⩾ β, by putting Γ(β, β, Ω) = Ω and extending Γ(n - 1, β, Ω) with a partial geometry.