The Geometry And Topology Of Coxeter Groups. (lms-32) (london Mathematical Society Monographs)
by Michael W. Davis /
2007 / English / PDF
4.3 MB Download
The Geometry and Topology of Coxeter Groups
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.
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.