Talk:Geometric group theory

Examples
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)

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. --C S (Talk) 03:20, 12 April 2006 (UTC)

More development needed, and wanted
This entry may no longer be considered a stub, but much more work is needed. We should have:
 * a history section
 * an important results section much more extensive than the set of results currently on the page
 * most importantly, a references section! One of the most common questions I am asked is for a book or article `on geometric group theory'.  Although there may be no standard reference, a list should be generated.

Also, although Gromov's program is of course central to geometric group theory, it is not the only theme of geometric group theory. Other major avenues of research should be mentioned.

Wikipedia has the potential to be a powerful reference for geometric group theory (or any field). One idea for a geometric group theory class would be to cover the desired material, then have each student choose a topic about which to give a short presentation and create/augment a corresponding wikipedia page.

--169.237.30.150 21:54, 21 September 2007 (UTC)

I've been intending to work on this article at some point, and I'd be happy to collaborate with you on improving it. By the way, I really like your idea on how to run a graduate class&mdash;sort of a modern variant on the Moore method! :-) Jim 22:03, 21 September 2007 (UTC)

Major rewrite
I just did a major rewrite of the entire article, adding a history section, a list of important topics, lots of references etc. Probably more references would have to be added, but I am getting kind of tired. Comments are welcome. Regards, Nsk92 (talk) 20:07, 23 March 2008 (UTC)

Systolic groups
Systolic groups should make it into this article sooner or later. Katzmik (talk) 09:04, 27 October 2008 (UTC)

Name dropping
The excessive name dropping and authority worship makes this article painful to read. I suggest revision of the style. I'm aware that this sort of terrible writing exists in science literature but it's not an excuse to go down that path here. — Preceding unsigned comment added by 129.100.144.108 (talk) 17:22, 3 July 2012 (UTC)

Dashes
I found more than 50 page ranges that used a hyphen rather than an en-dash, thus
 * 145-182

instead of
 * 145–182

There was not one instance in which it was done correctly. Also Baum-Connes appeared instead of Baum–Connes.

All corrected per WP:MOS. Michael Hardy (talk) 11:46, 3 June 2013 (UTC)

Finitely generated?
It was not my understanding that Geometric Group Theory is entirely confined to the study of finitely generated groups. We can use the rose with infinitely many petals and its covering spaces to understand the free group on infinitely many generators. We can use infinite coverings of finite roses to study infinitely generated subgroups of finite-rank free groups. — Preceding unsigned comment added by Davyker (talk • contribs) 21:00, 10 September 2018 (UTC)

CAT(0) cubical complexes
A page on CAT(0) cubical complexes should be added as those are an important part of geometric group theory DocLara (talk) 13:37, 7 April 2024 (UTC)