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On irreducible, infinite, non-affine coxeter groups

Qi, Dongwen
http://rave.ohiolink.edu/etdc/view?acc_num=osu1185463175

Abstract Details

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)
Mathematics
root system; canonical representations; irreducible Coxeter groups; parabolic subgroup; essential element; CAT(0) space; flat torus theorem; solvable subgroup theorem

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Citations

  • Qi, D. (2007). On irreducible, infinite, non-affine coxeter groups [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1185463175

    APA Style (7th edition)

  • Qi, Dongwen. On irreducible, infinite, non-affine coxeter groups. 2007. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1185463175.

    MLA Style (8th edition)

  • Qi, Dongwen. "On irreducible, infinite, non-affine coxeter groups." Doctoral dissertation, Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc_num=osu1185463175

    Chicago Manual of Style (17th edition)

osu1185463175
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© 2007, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.