Spreads in $PG(3,q)$ admitting several homology groups of order $q+1$

Abstract

The set of translation planes with spreads in $PG(3,q)$ admitting at least three homology groups with distinct axes of order $q+1$ is completely determined. Apart from the Desarguesian and Hall planes of order $q²$, the only possible plane is the Heimbeck plane of order $7²$ admitting several quaternion homology groups of order 8. A classification is also given of all translation planes with spreads in $PG(3,q)$ that admit at least three distinct homology groups of order $q+1$. Recent results conneting translation planes with spreads in $PG(3,q)$ admitting cyclic affine homology groups of order $q+1$ with conical flocks spreads provide the background for applications showing how the associated collineation groups are interrelated.