On homomorphisms between generalized polygons
The first result is that a non-injective homomorphism between generalized quadrangles necessarily has infinite fibers. This implies the result of \textit{A. Pasini} [ibid. 15, 389-397 (1984; Zbl 0537.51002)]\ starting that such homomorphisms do not exist on finite polygons. The proof employs the algebraic coordinate systems of \textit{Th. Grundhöfer}, \textit{N. Knarr} and the author [ibid. 55, No. 1, 95-114 (1995)]. Along the way, it is shown that non-injectivity can always be detected on any given line pencil or point row. In topological polygons, the arguments are carried further to show that a continuous non-injective homomorphism is even locally constant, whence Pasini's theorem generalizes to connected polygons. For polygons that are smooth (of class \(C^k\) with \(k\geq 2\)), it is proved next that each continuous homomorphism is a smooth embedding. It follows that the group of continuous automorphisms acts as a smooth transformation group. This is applied to prove that some of the isoparametric hypersurfaces \(F\) constructed by \textit{D. Ferus}, \textit{H. Karcher} and \textit{H. F. Münzner} [Math. Z. 177, 479-502 (1981; Zbl 0452.53032)]\ are inhomogeneous in the following sense. According to \textit{G. Thorbergsson} [Duke Math. J. 67, No. 3, 627-632 (1992; Zbl 0765.51015)], \(F\) is the flag space of a smooth polygon whose point and line sets are the focal manifolds of \(F\). If the line pencils have dimension \(m\equiv 3(4)\), then Ferus, Karcher and Münzner have shown that the space spanned by the tangent spaces of all point rows through a given point \(p\) has a dimension truly depending on \(p\). Therefore, the (smooth) automorphism group can act transitively neither on the point set, nor, a fortiori, on \(F\). This group may be larger than the isometry group of \(F\).
scientific article; zbMATH DE number 806293
scientific article