User:Cullinane/Translation plane

Definition

From P. J. Cameron on_projective_planes (pdf):

In a projective plane,

"Let p be a point and L a line. A central collineation with centre p and axis L is a collineation fixing every point on L and every line through p. It is called an elation if p is on L, a homology otherwise. The central collineations with centre p and axis L form a group."

From the site_on_geometry of H._Klein:

"A projective plane pi is called a translation plane if there exists a line l such that the group of elations with axis l is transitive on the affine plane pi_l [the "affine derivative" of pi]."

Relationship to spreads

Translation planes are related to spreads in projective spaces by the André/Bruck-Bose construction.

From Flocks,_ovals,_and_generalized_quadrangles (ps), (Four lectures in Napoli, June 2000), by Maska Law and Tim Penttila):

"A spread of PG(3,q) is a set of q²+1 lines, no 2 intersecting. (Equivalently, it is a partition of the points of PG(3,q) into lines.)"

"Given a spread S of PG(3,q), the André/Bruck-Bose construction¹ produces a translation plane pi(S) of order q² as follows: Embed PG(3,q) as a hyperplane of PG(4,q). Define an incidence structure A(S) with points the points of PG(4,q) not on PG(3,q) and lines the planes of PG(4,q) meeting PG(3,q) in a line of S. Then A(S) is a translation affine plane of order q². Let pi(S) be the projective completion of A(S)."

¹ See

• Johannes André, Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe, Math Z. 60, pp. 156-186, 1954, and
• R. H. Bruck and R. C. Bose, The construction of translation planes from projective spaces, J. Algebra 1, pp. 85-102, 1964.

Related reading

• Various publications_of_Keith_E._Mellinger (2001-2004) detail the close relationship between finite translation planes and spreads.

• See Foundations_of_Translation_Planes (2001), by M. Biliotti, V. Jha, and N. L. Johnson, for an extensive treatment of how spreads and translation planes are related.