Translation planes admitting many homologies (Q1321696)

There are a large number of examples of translation planes of odd order \(q^ 2\) admitting more than \(q+1\) axes of nontrivial homology groups of a given order. For translation planes of even nonsquare order or for odd order and odd dimension, a result of \textit{T. G. Ostrom} [Proc. Lond. Math. Soc., III. Ser. 26, 605-629 (1973; Zbl 0257.50009)] says that if there are ``many'' homology axes and if there are no elations, then any homology group of prime order must have order less than 5. In the paper under review the authors prove the following result. Theorem: Let \(\pi\) be a translation plane of even order \(q^ 2\) with kernel containing \(K\cong \text{GF}(q)\) that admits more than \(q+1\) distinct homology axes (or coaxes) of nontrivial affine homology groups with the same order. Then either 1) all these homology groups have the same axis or all have the same coaxis, and there is an affine elation group of order greater than \(q+1\), 2) \(\pi\) is a Hall plane, or 3) \(\pi\) is the Ott-Schaeffer plane of order \(2^{2r}\), where \(r\) is odd and the order of the homology groups is 3. Using this result the authors show that for any translation plane of even order \(q^ 2\) with kernel containing \(K\cong \text{GF} (q)\) which admits an homology group of order \(q+1\), either the axis and coaxis of the homology group are both fixed by the full collineation group or the plane is an André plane. Finally, the authors also look at the higher dimensional setting. Let \(\pi\) be a translation plane of (even or odd) order \(q^ n\) with kernel containing \(K\cong \text{GF}(q)\), and assume that \(\pi\) admits at least two cyclic homology groups of order \((q^ n-1)/ (q-1)\). If \(n>2\), it is shown that \(\pi\) must be an André plane.