Epimorphisms of Generalized Polygons, Part 2: Some Existence and Non-Existence Results

Gramlich, Ralf; Maldeghem, Hendrik Van

doi:10.1007/978-1-4613-0283-4_12

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…This implies that epimorphisms between finite generalized n-gons are always isomorphisms. Epimorphisms from a generalized n-gon to a generalized m-gon with m < n are studied by Gramlich and Van Maldeghem in [10] and [11].…”

Section: Known Results On Epimorphisms Of Spherical Buildingsmentioning

confidence: 99%

On epimorphisms of spherical Moufang buildings

Struyve

2013

Advances in Mathematics

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In this paper we classify the the epimorphisms of irreducible spherical Moufang buildings (of rank ≥ 2) defined over a field. As an application we characterize indecomposable epimorphisms of these buildings as those epimorphisms arising from R-buildings.1 of Moufang buildings defined over a field (made precise in Section 2.3.3) correspond to valuations of the underlying field satisfying certain compatibility conditions. Epimorphisms arising from Moufang R-buildings turn out to be the 'primitive' epimorphisms for this class, i.e. if one cannot decompose the epimorphism into two proper epimorphisms, then the epimorphism arises directly from an R-building.For a precise version of the main results and corollaries we refer to Section 3.This extends known results for projective spaces (see Section 2.5). The remaining open class, consisting of certain polar spaces of pseudo-quadratic form type defined over a nonabelian skew field is handled by the author and Petra N. Schwer in a forthcoming paper ([20]) using different, case-specific methods.Acknowledgement. The author would like to thank Pierre-Emmanuel Caprace for suggesting the problem. Preliminaries BuildingsLet (W, S) be a Coxeter system, then a weak building of type (W, S) is a pair (C, δ) consisting of a nonempty set C (called chambers) and a map δ : C × C → W (called the Weyl distance), such that for every two chambers C and D the following holds. (WD1) δ(C, D) = 1 if and only if C = D. (WD2) If δ(C, D) = w and C ′ ∈ C satisfies δ(C ′ , C) = s ∈ S, then δ(C ′ , D) ∈ {sw, w}. If moreover l(sw) = l(w) + 1 (where l is the word metric on W w.r.t. S), then δ(C ′ , D) = sw. (WD3) If δ(C, D) = w, then for any s ∈ S there exists a chamber C ′ ∈ C such that δ ′ (C ′ , C) = s and δ(C ′ , D) = sw.This weak building is said to be spherical if the Coxeter group W is finite. The rank of a weak building is defined to be |S|. Two chambers are s-equivalent (with s ∈ S) if the Weyl distance between them is either s or the identity element 1 of W . Consider a subset S ′ ⊂ S. The connected components of C using only equivalences in S ′ are called the S ′ -residues,

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Section: Known Results On Epimorphisms Of Spherical Buildingsmentioning

confidence: 99%

On epimorphisms of spherical Moufang buildings

Struyve

2013

Advances in Mathematics

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“…We say that a morphism α is an epimorphism if α(P) = P ′ and α(L) = L ′ . Contrary to Gramlich and Van Maldeghem [3,4], we do not ask surjectivity onto the set of flags of Γ ′ (the incident point-line pairs of Γ ′ ).…”

Section: Morphisms and Epimorphismsmentioning

confidence: 99%

“…A local variation on Theorem 1.1 by Bödi and Kramer [1] states that an epimorphism between thick generalized m-gons (m ≥ 3) is an isomorphism if and only if its restriction to at least one point row or line pencil is bijective. Later on, Gramlich and Van Maldeghem thoroughly studied epimorphisms from thick generalized m-gons to thick generalized n-gons with n < m in their works [3,4], and again, classification results are obtained based on the local nature of the epimorphisms. Finally, we mention recent work of Koen Struyve [11] on epimorphisms of spherical Moufang buildings.…”

Section: Introductionmentioning

confidence: 99%

Epimorphisms of generalized polygons A: The planes, quadrangles and hexagons

Thas,

Thas

2022

Preprint

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Inspired by a theorem by 7,8] and a problem which turned up in our recent paper [13], we start a study of epimorphisms with source a thick generalized m-gon and target a thin generalized m-gon. In this first part of the series, we classify the cases m = 3, 4 and 6 when the polygons are finite. Then we show that the infinite case is very different, and construct examples which strongly deviate from the finite case. A number of general structure theorems are also obtained. We introduce the theory of locally finitely generated generalized polygons and locally finitely chained generalized polygons along the way.

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“…A local variation on Theorem 1.1 by Bödi and Kramer [1] states that an epimorphism between thick generalized m-gons (m ≥ 3) is an isomorphism if and only if its restriction to at least one point row or line pencil is bijective. Later on, Gramlich and Van Maldeghem thoroughly studied epimorphisms from thick generalized m-gons to thick generalized n-gons with n < m in their works [4,5], and again, classification results were obtained based on the local nature of the epimorphisms.…”

Section: Introductionmentioning

confidence: 99%

Epimorphisms of generalized polygons, B: The octagons

Thas,

Thas

2023

Innov. Incidence Geom.

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This is the second part of our study of epimorphisms with source a thick generalized m-gon and target a thin generalized m-gon. We classify the case m = 8 when the polygons are finite (in the first part [15] we handled the cases m = 3, 4 and 6). Then we show that the infinite case is very different, and construct examples which strongly differ from the finite case. A number of general structure theorems are also obtained, and we also take a look at the infinite case for general gonality. CONTENTS 1. Introduction 1 2. Some basic definitions 2 3. Synopsis of known results 3 4. Epimorphisms to thin octagons 4 5. Locally finitely chained and generated generalized polygons 9 6. Counter examples in the infinite case 11 References 13

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Epimorphisms of Generalized Polygons, Part 2: Some Existence and Non-Existence Results

Cited by 8 publications

References 10 publications

On epimorphisms of spherical Moufang buildings

On epimorphisms of spherical Moufang buildings

Epimorphisms of generalized polygons A: The planes, quadrangles and hexagons

Epimorphisms of generalized polygons, B: The octagons

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