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University of Oklahoma
1.
Tucker, Cherith Anne.
Geodesic fibrations of elliptic 3-manifolds.
Degree: PhD, 2013, University of Oklahoma
URL: http://hdl.handle.net/11244/319012 
► The well-known Hopf fibration of S3 is interesting in part because its fibers are geodesics, or great circles, of S3. However, this is not the…
(more)
▼ The well-known Hopf fibration of S3 is interesting in part because its fibers are geodesics, or great circles, of S3. However, this is not the only great circle fibration of S3. In 1983, Herman Gluck and Frank Warner used the fact that the space of all oriented geodesics of the 3-sphere is homeomorphic to S2 × S2 to establish that there are many other great circle fibrations of S3. They showed that a submanifold of S2 × S2 corresponds to a fibration of S3 by oriented great circles if and only if it is the graph of a distance decreasing map from either S2 factor to the other. Since S3 is the universal cover of all elliptic 3-manifolds, we use this result to investigate geodesic Seifert fibrations of elliptic 3-manifolds. We also develop a different perspective on the space of oriented geodesics in S3 than that used by Gluck and Warner, and we examine its role in studying the geometry of the 3-sphere.
Advisors/Committee Members: Miller, Andrew (advisor).
Subjects/Keywords: Geodesics (Mathematics); Topology; Three-manifolds (Topology)
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APA (6th Edition):
Tucker, C. A. (2013). Geodesic fibrations of elliptic 3-manifolds. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/319012
Chicago Manual of Style (16th Edition):
Tucker, Cherith Anne. “Geodesic fibrations of elliptic 3-manifolds.” 2013. Doctoral Dissertation, University of Oklahoma. Accessed March 03, 2021.
http://hdl.handle.net/11244/319012.
MLA Handbook (7th Edition):
Tucker, Cherith Anne. “Geodesic fibrations of elliptic 3-manifolds.” 2013. Web. 03 Mar 2021.
Vancouver:
Tucker CA. Geodesic fibrations of elliptic 3-manifolds. [Internet] [Doctoral dissertation]. University of Oklahoma; 2013. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/11244/319012.
Council of Science Editors:
Tucker CA. Geodesic fibrations of elliptic 3-manifolds. [Doctoral Dissertation]. University of Oklahoma; 2013. Available from: http://hdl.handle.net/11244/319012
University of Oklahoma
2.
Wright, Rachel.
Totally Reflected Groups.
Degree: PhD, 2016, University of Oklahoma
URL: http://hdl.handle.net/11244/34633
► A group G is totally reflected if it has a generating set S such that each edge in the Cayley graph Gamma(G,S) is inverted by…
(more)
▼ A group G is totally reflected if it has a generating set S such that each edge in the Cayley graph Gamma(G,S) is inverted by some color-preserving reflection on the graph. For example, we will show that Coxeter groups and right-angled Artin groups are totally reflected and that a finitely generated abelian group is totally reflected if and only if its first invariant factor is even. We show that direct and free products of totally reflected groups are totally reflected. More generally, we develop a group construction called a right-angled product which generalizes free and direct products, and we show that a right-angled product of totally reflected groups is itself totally reflected.
A group G is strongly totally reflected if there exists a color-preserving reflection group G_R acting on Gamma(G,S) such that each edge in the graph is inverted by some reflection in G_R. We state and prove sufficient conditions for a totally reflected group to be strongly totally reflected and use these results to prove from a graphical perspective that any right-angled Artin group is commensurable with a right-angled Coxeter group. In particular, we show that both the right-angled Artin group A(Delta)=<S> and its associated right-angled Coxeter group A_r are finite-index subgroups of the group of color-preserving graph automorphisms of Gamma(A(Delta),S).
Advisors/Committee Members: Miller, Andrew (advisor), Rubin, Leonard (committee member), Reeder, Stacy (committee member), Stewart, Sepideh (committee member), Ozaydin, Murad (committee member).
Subjects/Keywords: Mathematics.; graph reflections; right-angled product; Cayley graph; geometric group theory
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APA (6th Edition):
Wright, R. (2016). Totally Reflected Groups. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/34633
Chicago Manual of Style (16th Edition):
Wright, Rachel. “Totally Reflected Groups.” 2016. Doctoral Dissertation, University of Oklahoma. Accessed March 03, 2021.
http://hdl.handle.net/11244/34633.
MLA Handbook (7th Edition):
Wright, Rachel. “Totally Reflected Groups.” 2016. Web. 03 Mar 2021.
Vancouver:
Wright R. Totally Reflected Groups. [Internet] [Doctoral dissertation]. University of Oklahoma; 2016. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/11244/34633.
Council of Science Editors:
Wright R. Totally Reflected Groups. [Doctoral Dissertation]. University of Oklahoma; 2016. Available from: http://hdl.handle.net/11244/34633
University of Oklahoma
3.
Carter, William.
Dehn Functions of the Stallings-Bieri Groups and Constructions of Non-Unique Product Groups.
Degree: PhD, 2015, University of Oklahoma
URL: http://hdl.handle.net/11244/14584
► This thesis will consist of two separate halves in which we will present results concerning two different families of finitely generated torsion-free groups. The themes…
(more)
▼ This thesis will consist of two separate halves in which we will present results concerning two different families of finitely generated torsion-free groups. The themes of each half are quite different and are related to certain geometric and non-geometric properties of groups. Considerations for both types of properties allow us to better understand the algebraic structure of such groups.
In the first half of the thesis, we examine the Dehn functions of a family of subgroups of right-angled Artin groups. If G is a finitely generated group, then the Dehn function gives the optimal isoperimetric function of a simply connected 2-complex that is quasi-isometric to the Cayley graph of G. In the first half of this thesis, we will show that if a graph Γ can be decomposed as a non-trivial join of three smaller subgraphs, then the Bestvina-Brady group, B_Γ, has a quadratic Dehn function. This result proves that the Stallings-Bieri groups SB
n have quadratic Dehn functions, for n ≥ 3, establishing a claim made by Bridson.
The second half of this thesis is motivated by the study of two well known conjectures in the theory of group rings over torsion-free groups, namely Kaplansky's Zero Divisor Conjecture and Non-trivial Units Conjecture. Together, they represent basic information we would like to know about any given group ring. It is known that if a group satisfies the unique product property, then any group ring over this group also satisfies both conjectures, and so torsion-free groups that do not satisfy this property are likely candidates for counter examples. Unfortunately, very little is known about torsion-free groups that do not have this property. Specifically, there are only two known examples of such groups, a single group with an explicit presentation, and a family of groups produced via a complex procedure. From these two examples, one can trivially produce infinitely many examples via products and embeddings; however, it is currently unclear how to produce genuinely new examples of such groups. Currently, to demonstrate that such a group does not satisfy this property requires producing two finite sets whose product has no uniquely represented element. We will refer to such sets as a pair of non-unique product sets. In the second part of this thesis, we construct a new non-trivial family of examples with explicit presentations and show that these groups contain arbitrarily large non-unique product sets.
Advisors/Committee Members: Forester, Max (advisor), Brady, Noel (committee member), Miller, Andrew (committee member), Kujawa, Jonathan (committee member), Hale, Piers (committee member).
Subjects/Keywords: Mathematics. Geometric Group Theory. Group Theory.
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APA ·
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MLA ·
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APA (6th Edition):
Carter, W. (2015). Dehn Functions of the Stallings-Bieri Groups and Constructions of Non-Unique Product Groups. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/14584
Chicago Manual of Style (16th Edition):
Carter, William. “Dehn Functions of the Stallings-Bieri Groups and Constructions of Non-Unique Product Groups.” 2015. Doctoral Dissertation, University of Oklahoma. Accessed March 03, 2021.
http://hdl.handle.net/11244/14584.
MLA Handbook (7th Edition):
Carter, William. “Dehn Functions of the Stallings-Bieri Groups and Constructions of Non-Unique Product Groups.” 2015. Web. 03 Mar 2021.
Vancouver:
Carter W. Dehn Functions of the Stallings-Bieri Groups and Constructions of Non-Unique Product Groups. [Internet] [Doctoral dissertation]. University of Oklahoma; 2015. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/11244/14584.
Council of Science Editors:
Carter W. Dehn Functions of the Stallings-Bieri Groups and Constructions of Non-Unique Product Groups. [Doctoral Dissertation]. University of Oklahoma; 2015. Available from: http://hdl.handle.net/11244/14584
University of Oklahoma
4.
Yamamoto, Tetsuya.
Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction.
Degree: PhD, 2015, University of Oklahoma
URL: http://hdl.handle.net/11244/14637
► Studies have shown that proof construction is a challenging task for students at all levels. The purposes of my study were to examine students' difficulties…
(more)
▼ Studies have shown that proof construction is a challenging task for students at all levels. The purposes of my study were to examine students' difficulties with proof construction and to explore a practical method to help them overcome their difficulties. I created a model of the structure of proof construction. The model was useful in explaining students' difficulties and producing metacognitive knowledge for proof construction in the form of algorithm.
Advisors/Committee Members: Stewart, Sepideh (advisor), Albert, John (committee member), Kyung-Bai, Lee (committee member), Miller, Andrew (committee member), Savic, Milos (committee member), Terry, Robert (committee member).
Subjects/Keywords: Mathematics.
Record Details
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Record Details
Similar Records
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Yamamoto, T. (2015). Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction. (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/14637
Chicago Manual of Style (16th Edition):
Yamamoto, Tetsuya. “Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction.” 2015. Doctoral Dissertation, University of Oklahoma. Accessed March 03, 2021.
http://hdl.handle.net/11244/14637.
MLA Handbook (7th Edition):
Yamamoto, Tetsuya. “Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction.” 2015. Web. 03 Mar 2021.
Vancouver:
Yamamoto T. Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction. [Internet] [Doctoral dissertation]. University of Oklahoma; 2015. [cited 2021 Mar 03].
Available from: http://hdl.handle.net/11244/14637.
Council of Science Editors:
Yamamoto T. Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction. [Doctoral Dissertation]. University of Oklahoma; 2015. Available from: http://hdl.handle.net/11244/14637
. 
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