From Groups to Geometry and Back

TAILLE DU FICHIER: 10,16 MB

AUTEUR: Vaughn Climenhaga

ISBN: 9781470434793

NOM DE FICHIER: From Groups to Geometry and Back.pdf

DATE DE PUBLICATION: 2017-Jan-01

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Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes ; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth. The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness. This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009.
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PDF Livre From Groups to Geometry and Back

CORRECTIONS TO \FROM GROUPS TO GEOMETRY AND BACK" VAUGHN CLIMENHAGA I am grateful to Lewis Robinson for alerting me to the following errors. Page 87: In Figure 2.7, the labels d 2 and d 3 in the left-hand picture should be exchanged. Page 160: At the end of the second-to-last paragraph, the charac-teristic polynomial of the rotation matrix is missing a factor of in its second term. The correct ...

From Groups to Geometry and Back de Vaughn Climenhaga ...

From Groups to Geometry and Back. From AMS website. Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing ...