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On irreducible, infinite, non-affine coxeter groups
Qi, Dongwen

2007, Doctor of Philosophy, Ohio State University, Mathematics.
Coxeter groups arise in many parts of mathematics as groups generated by reflections. They provide an important source of examples in geometric group theory, where "virtual" properties of infinite groups, that is, properties of subgroups of finite index, are of strong interest. This dissertation focuses on virtual properties of infinite Coxeter groups. The following results are proved: (1) The intersection of a collection of parabolic subgroups of a Coxeter group is a parabolic subgroup; (2) The center of any finite index subgroup of an irreducible, infinite, non-affine Coxeter group is trivial; (3) Any finite index subgroup of an irreducible, infinite, non-affine Coxeter group cannot be expressed as a product of two nontrivial subgroups. Then, a unique decomposition theorem for subgroups of finite index in a Coxeter group without spherical or affine factors is proved based on (2) and (3). It is also proved that the orbit of each element other than the identity under the conjugation action in an irreducible, infinite, non-affine Coxeter group is an infinite set, which implies that an irreducible, infinite Coxeter group is affine if and only if it contains an abelian subgroup of finite index. Besides these, new proofs are given for the statement that the center of an irreducible, infinite Coxeter group is trivial, and a stronger version of this statement.
Michael Davis (Advisor)

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Qi, D. (2007). On irreducible, infinite, non-affine coxeter groups. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Qi, Dongwen. "On irreducible, infinite, non-affine coxeter groups." Electronic Thesis or Dissertation. Ohio State University, 2007. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Qi, Dongwen "On irreducible, infinite, non-affine coxeter groups." Electronic Thesis or Dissertation. Ohio State University, 2007. https://etd.ohiolink.edu/

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