on irreducible, infinite, non-affine coxeter groups

62 Pages·2010·0.47 MB·English

by  Michael Davis

ON IRREDUCIBLE, INFINITE, NON-AFFINE
COXETER GROUPS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Dongwen Qi, B.S., M.S.
* * * * *
The Ohio State University
2007
Dissertation Committee: Approved by
Professor Michael Davis, Advisor
Professor Jean-Francois Lafont
Advisor
Professor Ian Leary
Graduate Program in
Mathematics
Professor Fangyang Zheng
c Copyright by
!
Dongwen Qi
2007
ABSTRACT
Coxeter groups arise in many parts of mathematics as groups generated by re-
flections. 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 irre-
ducible, infinite Coxeter group is trivial, and a stronger version of this statement.
MSC 2000 Subject Classifications: Primary 20F55; Secondary 20F65, 57M07,
53C23
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Key Words: root system, canonical representations, irreducible Coxeter groups,
parabolic subgroup, essential element, CAT(0) space, flat torus theorem, solvable
subgroup theorem
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To my family
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ACKNOWLEDGMENTS
I am indebted to Professor Michael Davis and Professor Ruth Charney for direct-
ing me to the field of geometric group theory. I thank Professor Davis and Professor
Tadeusz Januszkiewicz for the inspiration and numerous discussion during the prepa-
ration of this dissertation. I would also like to thank the members of my disserta-
tion committee, Professor Jean-Francois Lafont, Professor Ian Leary and Professor
Fangyang Zheng for their invaluable time and discussion. I thank Professor Gordon
Aubrechtforhiscommentsandsuggestionsonadraftofmydissertation, whichIhave
incorporated in this version. I acknowledge the continuous support from the Math-
ematics Department at Ohio State for the past seven years and the partial support
from my advisor Michael Davis’ NSF grant.
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VITA
1988 ........................................B.S. in Mathematics
Beijing University, China
1991 ........................................M.S. in Mathematics
Beijing University, China
1995-1999 .................................. Lecturer, Department of Mechanics
and Engineering Science, Beijing Uni-
versity
1998-1999 .................................. Research Associate, City University of
Hong Kong
2000-present ................................Graduate Teaching Associate
The Ohio State University
PUBLICATIONS
Qi Dongwen, On irreducible, infinite, non-affine Coxeter groups, Fund. Math. 193
•
(2007), 79-93.
Qi Dongwen, A note on parabolic subgroups of a Coxeter group, Expo. Math. 25
•
(2007), 77-81.
Chan H.S.Y., Chung K.W., & Qi Dongwen Some bifurcation diagrams for limit
•
cycles of quadratic differential systems, Int. J. Bifurcation and Chaos 11 (2001)
197-206.
Chan H.S.Y., Chung K.W., & Qi Dongwen, Bifurcating limit cycles in quadratic
•
polynomial differential systems, Physica A 288 (2000), 417-423.
Qi Dongwen, & Jing Zhujun, Bifurcations of a pair of nonorientable heteroclinic
•
cycles, J. Math. Anal. Appl. 222 (1998), 319-338.
Qi Dongwen, Bifurcations of a pair of singular cycles, Nonlinear Anal., T.M.A. 29
•
(1997), 313-325.
Qi Dongwen, On a class of nonlocal bifurcation concerning the Lorenz attractor,
•
Acta Math. Sinica (N. S.) 12 (1996), 54-70.
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QiDongwen,Envelopingsemigroupandproximityrelationsintopologicaldynamics,
•
Advances in Math. (China) 25 (1996), 321-327.
FIELDS OF STUDY
Major Field: Mathematics
Studies in:
Topic 1 Geometric Group Theory, Metric Spaces of Nonpositive Curvature
Topic 2 Coxeter Groups
Topic 3 Bifurcations of Vector Fields
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TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Chapters:
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Basic Combinatorial Theory of Coxeter Groups . . . . . . . . . . . . . . 9
3. Canonical Representations of a Coxeter Group . . . . . . . . . . . . . . . 14
3.1 Canonical representations and root systems of a Coxeter group . . 14
3.2 The interior U0 of the Tits cone U . . . . . . . . . . . . . . . . . . 17
3.3 Root system of a parabolic subgroup and the parabolic closure of a
set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4. Some Theorems of Krammer and Paris . . . . . . . . . . . . . . . . . . . 25
4.1 Periodic, even and odd roots . . . . . . . . . . . . . . . . . . . . . 25
4.2 A pseudometric on U0 and the axis Q(w) . . . . . . . . . . . . . . 27
4.3 Critical roots and essential elements of a Coxeter group . . . . . . . 30
4.4 Proof of Theorem 4.17 on irreducible, infinite, non-affine Coxeter
groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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5. Proofs of The Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1 Proofs of Theorem 1.1 and 1.2 . . . . . . . . . . . . . . . . . . . . . 41
5.2 Proof of Theorem 1.3 and examples . . . . . . . . . . . . . . . . . . 45
5.3 Proof of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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