On the dimensions of automorphism groups of four-dimensional double loops
In order to get started on Salzmann's classification program for compact projective planes, one needs stiffness results on the automorphism group, i.e., upper bounds on the dimensions of the stabilizers of suitable configurations. One hopes, for example, that the stabilizer of a quadrangle (which coincides with the automorphism group of a coordinatizing ternary) cannot be bigger than in the classical plane of the same dimension. This has been proved for ternaries of all possible finite dimensions except 4, where \(3 = \dim \text{Aut }\mathbb{H}\) is the desired bound, but only the bound 5 could be proved [\textit{H. Salzmann}, Geom. Dedicata 8, 139-161 (1979; Zbl 0465.51003)]. The lack of a sharp bound caused much additional work in the above-mentioned program. Now the author obtains the improved bound 4, and he does so under a weaker hypothesis; in fact, he does not use the ternary operation itself, but only addition and multiplication. Thus, he proves the bound for automorphism groups of 4-dimensional double loops. In addition to the methods used previously in this context, he employs Alexander duality in order to show that the group cannot be transitive on the complement of the fixed point set \(F\). Here, the possibility causing the trouble is that \(F\) might be 0-dimensional. Earlier, the author has shown that the complement of \(F\) is simply connected in this case [Arch. Math. 61, 291- 294 (1993; Zbl 0783.22001)]. This result is played off against Alexander duality. For 8-dimensional double loops, a similar result has been proved by the author [Monatsh. Math. 117, 1-16 (1994; Zbl 0796.22004)]. There, his bound is not quite as strong as the one known for ternaries.
scientific article; zbMATH DE number 567419
scientific article