The geometry of the word problem for finitely generated groups
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. T...
| Main Author: | Brady, Noel. |
|---|---|
| Other Authors: | Riley, Tim, 1976-, Short, Hamish., SpringerLink (Online service) |
| Format: | eBook |
| Language: | English |
| Published: |
Basel ; Boston :
Birkhauser,
©2007.
Basel ; Boston : [2007] |
| Physical Description: |
1 online resource (206 pages) : illustrations. |
| Series: |
Advanced courses in mathematics, CRM Barcelona.
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| Subjects: |
| Summary: |
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 toDehn 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; it is discussed how filling functions interact, and applications to nilpotent groups, hyperbolic groups and asymptotic cones are given. Many examples and open problems are included. |
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| Item Description: |
Includes bibliographical references. Foreword -- Diagrams and Groups -- Dehn Functions and Non-Positive Curvature -- Filling Functions. 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 toDehn 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; it is discussed how filling functions interact, and applications to nilpotent groups, hyperbolic groups and asymptotic cones are given. Many examples and open problems are included. English. University staff and students only. Requires University Computer Account login off-campus. |
| Physical Description: |
1 online resource (206 pages) : illustrations. |
| Bibliography: |
Includes bibliographical references. |
| ISBN: |
9783764379506 3764379502 9783764379490 3764379499 |
| Access: |
University staff and students only. Requires University Computer Account login off-campus. |