Talk:Geometric group theory

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Geometric group theory certainly has its source and motivation in many examples. To the examples proposed in the stub article, let me add

• Triangle reflection groups, and other groups acting on the sphere, Euclidean plane, and hyperbolic plane. Links to Fuchsian groups.

• • Wallpaper groups.

• • Various of M. C. Escher's prints.

• Dehn's algorithm for solving the word problem in the fundamental group of a hyperbolic surface, and the extension to the word problem for hyperbolic groups

• Kleinian groups, acting on hyperbolic three space

• Other lattices acting on symmetric spaces.

However, since the early 1980's there have developed important broad themes which bind together the study of these example, and which motivate the study of arbitrary finitely generated groups. So it is not accurate to say that geometric group theory is mainly the study of some particular examples.

One main theme that should be presented is Gromov's program of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry. This program involves:

• A description of properties that are invariant under quasi-isometry, for example

• • the growth rate of a finitely generated group

• • the isoperimetric function or Dehn function of a finitely generated group

• • ends of a group

• • hyperbolicity of a group, and the boundary of a hyperbolic group (this stub article contains a reference to quasi-isometry invariance of hyperbolicity)

• Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example

• • Gromov's polynomial growth theorem

• • Stallings ends theorem

• • Mostow's rigidity theorem.

• Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space.

This is a big theme, and I suspect that to develop it will require lots of new articles.

--Mosher 13:14, 24 September 2005 (UTC)

[edit] Moved from article

• What does it mean for a group to act on a space? What kinds of actions do we care about in geometric group theory?
• The Cayley graph as the canonical space to act on. The adjacency matrix of a Cayley graph allows number-theoretic methods to be applied as well, via spectral graph theory.
• The Ping-Pong lemma, which is the main way to exhibit a group as a free product
• Finiteness properties
• Amenability, as it is studied by geometric group theory

As Mosher points out, geometric group theory is not considered to be just the study of some examples. So I changed the list of examples somewhat; hopefully I didn't delete anything people object to. --Chan-Ho (Talk) 03:20, 12 April 2006 (UTC)