The Geometry of the Word Problem for Finitely Generated Groups (Advanced Courses in Mathematics - CRM Barcelona) - Softcover

Brady, Noel; Riley, Tim; Short, Hamish

Softcover

ISBN 10: 3764379499 ISBN 13: 9783764379490

Publisher: Birkhäuser, 2006

Synopsis

The origins of the word problem are in group theory, decidability and complexity. But through the vision of M. Gromov and the language of filling functions, the topic now impacts the world of large-scale geometry. This book contains accounts of many recent developments in Geometric Group Theory and shows the interaction between the word problem and geometry continues to be a central theme. It contains many figures, numerous exercises and open questions.

About this title

Synopsis

The origins of the word problem are in group theory, decidability and complexity, but, through the vision of M. Gromov and the language of filling functions, the topic now impacts the world of large-scale geometry, including topics such as soap films, isoperimetry, coarse invariants and curvature. The first part introduces van Kampen diagrams in Cayley graphs of finitely generated, infinite groups; it discusses the van Kampen lemma, the isoperimetric functions or Dehn functions, the theory of small cancellation groups and an introduction to hyperbolic groups. One of the main tools in geometric group theory is the study of spaces, in particular geodesic spaces and manifolds, such that the groups act upon. The second part is thus dedicated to Dehn functions, negatively curved groups, in particular, CAT(0) groups, cubings and cubical complexes. In the last part, filling functions are presented from geometric, algebraic and algorithmic points of view, how filling functions interact is discussed, and applications to nilpotent groups, hyperbolic groups and asymptotic cones are given. Many examples and open problems are included.

Edition details

• Publisher: Birkhäuser
• Publication date: 2006
• Language: English
• ISBN 10: 3764379499
• ISBN 13: 9783764379490
• Binding: Paperback
• Number of pages: 215