The Geometry of the Word Problem for Finitely Generated Groups / by Noel Brady, Tim Riley, Hamish Short.

Springer eBooks

Dehn Functions and Non-Positive Curvature -- The Isoperimetric Spectrum -- Dehn Functions of Subgroups of CAT(0) Groups -- Filling Functions -- Filling Functions -- Relationships Between Filling Functions -- Example: Nilpotent Groups -- Asymptotic Cones -- Diagrams and Groups -- Dehn’s Problems and Cayley Graphs -- Van Kampen Diagrams and Pictures -- Small Cancellation Conditions -- Isoperimetric Inequalities and Quasi-Isometries -- Free Nilpotent Groups -- Hyperbolic-by-free 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. 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; 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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