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Books > Science & Mathematics > Mathematics > Algebra
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From Groups to Geometry and Back (Paperback)
Loot Price: R1,590
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From Groups to Geometry and Back (Paperback)
Series: Student Mathematical Library
Expected to ship within 12 - 17 working days
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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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