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Metrically homogeneous graphs
Descriptive
TitleInfo
Title
Metrically homogeneous graphs
SubTitle
dynamical properties of their automorphism groups and the classification of twists
Name
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Coulson
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Rebecca
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Rebecca Coulson
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author
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Thomas
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Simon Rhys
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Simon Rhys Thomas
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Advisory Committee
Role
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chair
Name
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Cherlin
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Gregory
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Gregory Cherlin
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Advisory Committee
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internal member
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(type = personal)
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(type = family)
Sargsyan
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Grigor
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Grigor Sargsyan
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Advisory Committee
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internal member
Name
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Patel
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Rehana
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Rehana Patel
Affiliation
Advisory Committee
Role
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outside member
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Rutgers University
Role
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degree grantor
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School of Graduate Studies
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Text
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theses
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2019
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2019-01
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2019
Place
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xx
Language
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eng
Abstract
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We investigate the properties of graphs which are homogeneous in the sense of Fraisse when considered as metric spaces with the graph metric (metrically homogeneous graphs), and particularly the metrically homogeneous graphs of generic type constructed by Cherlin.
We first consider the properties of the associated automorphism groups, viewed as topological groups. For a large class of metrically homogeneous graphs of generic type, we show that the automorphism groups have ample generics, and therefore have a variety of topological properties such as the small index property and automatic continuity. We also show that the automorphism groups of the generic expansions of these graphs by linear orders are extremely amenable, and describe the universal minimal glow for the full automorphism group. Using standard model theoretic and descriptive set theoretic methods, these results are derived from the study of combinatorial properties of the associated classes of finite partial substructures.
Turning to more algebraic questions, we determine the twisted automorphism groups of metrically homogeneous graphs, and more generally the twisted isomorphisms between such graphs; these are isomorphisms up to a permutation of the natural language. Returning to the standard automorphism group, we then study the algebra of the associated age in the sense of Peter Cameron, showing that in most cases this algebra is a polynomial algebra. For this, we apply a criterion of Cameron based on a unique decomposition theorem.
We first consider the properties of the associated automorphism groups, viewed as topological groups. For a large class of metrically homogeneous graphs of generic type, we show that the automorphism groups have ample generics, and therefore have a variety of topological properties such as the small index property and automatic continuity. We also show that the automorphism groups of the generic expansions of these graphs by linear orders are extremely amenable, and describe the universal minimal glow for the full automorphism group. Using standard model theoretic and descriptive set theoretic methods, these results are derived from the study of combinatorial properties of the associated classes of finite partial substructures.
Turning to more algebraic questions, we determine the twisted automorphism groups of metrically homogeneous graphs, and more generally the twisted isomorphisms between such graphs; these are isomorphisms up to a permutation of the natural language. Returning to the standard automorphism group, we then study the algebra of the associated age in the sense of Peter Cameron, showing that in most cases this algebra is a polynomial algebra. For this, we apply a criterion of Cameron based on a unique decomposition theorem.
Subject
(authority = RUETD)
Topic
Mathematics
Subject
(authority = ETD-LCSH)
Topic
Automorphisms
Subject
(authority = ETD-LCSH)
Topic
Spaces of homogeneous type
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Rutgers University Electronic Theses and Dissertations
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ETD_9519
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electronic resource
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Extent
1 online resource (160 pages : illustrations)
Note
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Ph.D.
Note
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Includes bibliographical references
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by Rebecca Coulson
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School of Graduate Studies Electronic Theses and Dissertations
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rucore10001600001
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NjNbRU
Identifier
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doi:10.7282/t3-exe2-jg62
Genre
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ETD doctoral
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Rights
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The author owns the copyright to this work.
RightsHolder
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Name
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Coulson
GivenName
Rebecca
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Type
Permission or license
DateTime
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2019-01-09 22:11:53
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Name
Rebecca Coulson
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Affiliation
Rutgers University. School of Graduate Studies
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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