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On the local-global conjecture for commutator traces
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Ogrodnik, Brooke. On the local-global conjecture for commutator traces. Retrieved from https://doi.org/doi:10.7282/t3-xsab-0520

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TitleOn the local-global conjecture for commutator traces
NameOgrodnik, Brooke (author); Kontorovich, Alex (chair); Iwaniec, Henryk (internal member); Miller, Stephen D (internal member); Sarnak, Peter (outside member); Rutgers University; School of Graduate Studies
Date Created2021
Other Date2021-05 (degree)
SubjectThin groups, Mathematics
Extent1 online resource (viii, 86 pages)
DescriptionWe study the trace set of the commutator subgroup of Gamma(2), a type of Local-Global problem about thin groups. We determine the local obstructions and then use the correspondence between binary quadratic forms and hyperbolic matrices to find some global obstructions. We then develop a probabilistic argument for the existence of sufficiently large admissible traces by modeling the elements as non-backtracking random walks in a 2-dimensional lattice via their homology class and word length. Finally, we investigate the number of commutators needed to represent matrices in the commutator subgroup. This is done using an algorithm of Goldstein and Turner along with utilizing properties of level k-Markoff type surfaces. We conjecture that any trace in the commutator subgroup of $Gamma(2)$ can be represented with either 1 or 2 commutators.
NotePh.D.
NoteIncludes bibliographical references
Genretheses, ETD doctoral
Persistent URLhttps://doi.org/doi:10.7282/t3-xsab-0520
LanguageEnglish
CollectionSchool of Graduate Studies Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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