On homomorphisms between generalized polygons
type: Item
instance of: scholarly article
title: On homomorphisms between generalized polygons
zbMATH Open document ID: 0836.51001
DOI: 10.1007/BF01263471
author: Richard Bödi
author: Linus Kramer
published in: Geometriae Dedicata
publication date: 1996-04-28T00:00:00Z
review text: 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 \(F\).
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 51H15
zbMATH DE Number: 806293
zbMATH Keywords: generalized polygons
zbMATH Keywords: topological polygon
zbMATH Keywords: isoparametric hypersurface
zbMATH Keywords: homomorphisms
reviewed by: Rainer Löwen
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01263471
OpenAlex ID: W2038539321
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description: scientific article; zbMATH DE number 806293