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This dissertation by Mikael Hansson explores combinatorial aspects of Coxeter group theory, focusing on involutions and their topological properties. It comprises three papers that delve into the classification and structural analysis of posets derived from Coxeter groups. The work investigates the properties of involutions in symmetric and twisted forms, providing new proofs and extending existing theories. The research also examines special partial matchings in posets, known as pircons, and their topological characteristics, such as being PL balls or spheres. The dissertation is intended for mathematicians and researchers in algebra, geometry, and combinatorics.

Coxeter groups. --- Combinatorial topology.

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This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.

Coxeter groups. --- Reflection groups.

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Coxeter groups --- Geometry, Algebraic --- Group theory

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The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

512.54 --- 512.54 Groups. Group theory --- Groups. Group theory --- Coxeter groups --- Geometric group theory --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Group theory --- Coxeter groups. --- Geometric group theory.

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The book is the first to give a comprehensive overview of the techniques and tools currently being used in the study of combinatorial problems in Coxeter groups. It is self-contained, and accessible even to advanced undergraduate students of mathematics. The primary purpose of the book is to highlight approximations to the difficult isomorphism problem in Coxeter groups. A number of theorems relating to this problem are stated and proven. Most of the results addressed here concern conditions which can be seen as varying degrees of uniqueness of representations of Coxeter groups. Throughout the

Coxeter groups. --- Isomorphisms (Mathematics) --- Categories (Mathematics) --- Group theory --- Morphisms (Mathematics) --- Set theory --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real