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Centralisers of fundamental subgroups

Altmann, Kristina (2007)
Centralisers of fundamental subgroups.
Technische Universität Darmstadt
Ph.D. Thesis, Primary publication

Abstract

A central problem in synthetic geometry is the characterisation of graphs and geometries. The local recognition of locally homogeneous graphs forms one category of such characterisations, which works as follows: Let Δ be a graph. A graph Γ is called locally Δ if for each vertex x of Γ the graph Γx is isomorphic to Δ, where Γx is the induced subgraph of Γ on the set of vertices adjacent to x. It is a natural question to ask for all (connected) graphs, which are locally some graph Δ. This classification question is called the local recognition problem for graphs that are locally Δ, which can be found in great quantities in the literature. One of the earliest and most influential is [1]. The present thesis solves a number of open local recognition problems for different graphs and geometries. First we give an elementary and self-contained charaterization of connected graphs that are locally isomorphic to the line graph of an n-dimensional unitary vector space over the complex numbers endowed with an anisotropic form for n greater or equal to seven. Then we study the graph on the hyperbolic lines (non-degenerate lines) of a unitary vector space over a finite field of dimension at least seven, where two different hyperbolic lines are adjacent if and only if one hyperbolic line is in the polar space of the other hyperbolic line. We prove that the diameter of these graphs is two. This fact enables us to classify all these graphs by their internal properties via the theorem 1.2 of [2]. Finally a local recognation theorem for connected graphs, which are locally isomorphic to the line graph of the six-dimensional unitary vector space over the complex numbers endowed with an anisotropic form, is proved. We describe in the last part a method to construct some subgroup of the automorphism group of the considered graphs with certain properties for each vertex of the graph and its neighborhood. Moreover a classification of this subgroup of automorphisms is possible via a locally reflection graph and the theory of weak Phan systems. This classification of the constructed subgroup implies a local recognation of the universal cover of the graphs. [1] Francis Buekenhout, Xavier Hubaut, Locally polar spaces and related rank 3 groups, J. Algebra 45 (1977), 391--434 [2] Hans Cuypers, The geometry of hyperbolic lines in polar spaces, unpublished

Item Type: Ph.D. Thesis
Erschienen: 2007
Creators: Altmann, Kristina
Type of entry: Primary publication
Title: Centralisers of fundamental subgroups
Language: English
Referees: Joswig, Prof.-Dr. Michael ; Cuypers, Dr. Hans
Advisors: Gramlich, dr. Ralf
Date: 5 October 2007
Place of Publication: Darmstadt
Publisher: Technische Universität
Collation: XIV, 286 S. : graph. Darst.
Refereed: 6 February 2007
URL / URN: urn:nbn:de:tuda-tuprints-8750
Corresponding Links:
Abstract:

A central problem in synthetic geometry is the characterisation of graphs and geometries. The local recognition of locally homogeneous graphs forms one category of such characterisations, which works as follows: Let Δ be a graph. A graph Γ is called locally Δ if for each vertex x of Γ the graph Γx is isomorphic to Δ, where Γx is the induced subgraph of Γ on the set of vertices adjacent to x. It is a natural question to ask for all (connected) graphs, which are locally some graph Δ. This classification question is called the local recognition problem for graphs that are locally Δ, which can be found in great quantities in the literature. One of the earliest and most influential is [1]. The present thesis solves a number of open local recognition problems for different graphs and geometries. First we give an elementary and self-contained charaterization of connected graphs that are locally isomorphic to the line graph of an n-dimensional unitary vector space over the complex numbers endowed with an anisotropic form for n greater or equal to seven. Then we study the graph on the hyperbolic lines (non-degenerate lines) of a unitary vector space over a finite field of dimension at least seven, where two different hyperbolic lines are adjacent if and only if one hyperbolic line is in the polar space of the other hyperbolic line. We prove that the diameter of these graphs is two. This fact enables us to classify all these graphs by their internal properties via the theorem 1.2 of [2]. Finally a local recognation theorem for connected graphs, which are locally isomorphic to the line graph of the six-dimensional unitary vector space over the complex numbers endowed with an anisotropic form, is proved. We describe in the last part a method to construct some subgroup of the automorphism group of the considered graphs with certain properties for each vertex of the graph and its neighborhood. Moreover a classification of this subgroup of automorphisms is possible via a locally reflection graph and the theory of weak Phan systems. This classification of the constructed subgroup implies a local recognation of the universal cover of the graphs. [1] Francis Buekenhout, Xavier Hubaut, Locally polar spaces and related rank 3 groups, J. Algebra 45 (1977), 391--434 [2] Hans Cuypers, The geometry of hyperbolic lines in polar spaces, unpublished

Alternative Abstract:
Alternative abstract Language

Ein zentrales Problem in der synthetischen Geometrie ist die Charakterisierung von Graphen und Geometrien. Die lokale Erkennung von lokal homogenen Graphen bildet eine Kategorie solcher Charakterisierungen, welche wie folgt funktioniert: Sei Δ ein Graph. Ein Graph Γ ist lokal Δ, wenn für jede Ecke x von Γ der Graph Γx isomorph zu Δ ist, wobei Γx der induzierte Untergraph von Γ auf der Menge aller Ecken benachbart mit x ist. Es ist natürliche nach allen (zusammenhängenden) Graphen zu fragen, welche lokal Δ sind, wobei Δ irgendein gewählter Graph ist. Diese Fragestellung ist das lokale Erkennungsproblem von lokalen Δ Graphen, welches in großer Anzahl in der Literatur gefunden werden kann. Eins der ersten und einflußreichsten lokalen Erkennungsprobleme ist [1]. Diese Dissertation löst einige lokale Erkennungsprobleme für verschiedene Graphen und Geometrien. Zuerst geben wir eine elementare und in sich geschlossene Beschreibung von zusammenhängenden Graphen, welche lokal isomorph zum Geradengraphen eines n-dimensionalen unitären Vektorraumes über den komplexen Zahlen mit einer anisotropen Form für n größer gleich sieben. Dann studieren wir den Graphen auf den hyperbolischen Geraden (nicht degenerierten Geraden) des unitären Vektorraums über einen endlichen Körper von Dimension mindestens sieben, wobei genau dann zwei verschiedene hyperbolische Geraden benachbart sind, wenn eine hyperbolische Gerade im Senkrechtraum der anderen hyperbolischen Gerade ist und umgekehrt. Wir beweisen, daß der Durchmesser dieser Graphen zwei ist. Diese Tatsache ermöglicht die Klassifizierung all dieser lokalen Graphen durch ihre internen Eigenschaften und Satz 1.2 in [2]. Zuletzt wird ein lokaler Erkennungssatz für zusammenhängende Graphen, welche lokal isomorph zum Geradengraphen des sechs-dimensionalen unitären Vektorraumes über den komplexen Zahlen mit einer anisotropen Form sind, bewiesen. We geben eine Methode an um eine Untergruppe der Automorphismengruppe der Graphen mit bestimmten Eigenschaften für jede Ecke des Graphens und ihre Nachbarschaft, zu konstruieren. Des Weiteren ist eine Klassifikation dieser Untergruppe mittels einem lokalen Spiegelungsgraphen und der schwachen Phan-Theorie möglich. Diese Klassifizierung der konstruierten Untergruppe impliziert im letzten Abschnitt die lokale Erkennung der universellen Überlagerung der betrachteten Graphen. [1] Francis Buekenhout, Xavier Hubaut, Locally polar spaces and related rank 3 groups, J. Algebra 45 (1977), 391--434 [2] Hans Cuypers, The geometry of hyperbolic lines in polar spaces, unpublished

German
Uncontrolled Keywords: Inzidenzgeometrie, einfach zusammenhängender Graph, lokal homogener Graph, lokale Erkennung
Alternative keywords:
Alternative keywordsLanguage
incidence geometry, simple connected graph, local homogeneous graph, local recognationEnglish
Divisions: 04 Department of Mathematics
Date Deposited: 17 Oct 2008 09:22
Last Modified: 27 Jul 2023 11:19
PPN:
Referees: Joswig, Prof.-Dr. Michael ; Cuypers, Dr. Hans
Refereed / Verteidigung / mdl. Prüfung: 6 February 2007
Alternative keywords:
Alternative keywordsLanguage
incidence geometry, simple connected graph, local homogeneous graph, local recognationEnglish
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