Talk:Hyperbolic group

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Mathematics rating:	Start Class	Mid Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Arcfrk (talk) 14:51, 9 February 2008 (UTC)

Contents 1 Delta thin 2 Future directions 3 Computational Properties 4 Thin bigons 5 Illustrations

[edit] Delta thin

So is the definition that every geodesic triangle in the Cayley graph must be δ-thin for some δ ≥ 0? Do we have to fix a δ for which all geodesic triangles are δ-thin at once? This could be made more clear if this is the case. - Gauge 00:41, 19 August 2006 (UTC)

Presumably this is the case so I changed the wording accordingly, and put the definition of δ-hyperbolic in bold so it stands out. - Gauge 00:45, 19 August 2006 (UTC)

Yes, the definition is just that the Cayley graph is δ-hyperbolic. (Equivalently, the group itself with any word metric - the definition of δ-hyperbolicity does not require the space to be a length space. δ-hyperbolicity is preserved under quasi-isometries, since δ-hyperbolicity is equivalent to being quasi-geodesically stable (by the Morse Lemma)). But the definition about the Linear Isoperimetric Inequality is very important.

[edit] Future directions

Have to say that it doesn't matter which word metric is used, since a change of generators is a quasi-isometry.

Most importantly, a group is hyperbolic iff it satisfied a linear isoperimetric inequality. A combinatorial and a geometric interpretation is possible and a picture would be a good idea. A hyperbolic group is in fact, an automatic group, that is, multiplication in it can be checked by a finite automaton. 146.186.132.188 06:21, 14 October 2007 (UTC)

[edit] Computational Properties

I added a couple of sentences on the word problem for hyperbolic groups, and the fact that they are automatic. I'm pretty sure that there is actually a linear time solution to the word problem (the automation gives you a quadratic time one), but with large constants which make it impractical. I think this was proved by David Epstein, but it's some time since I looked at this stuff, and my books are on the other side of the world. If anyone can help, that would be much appreciated. TheAstonishingBadger 23:48, 9 November 2007 (UTC)

[edit] Thin bigons

One surprising fact about hyperbolic groups is that any group where bigons (two-sided polygons) are δ-thin are hyperbolic. That is, thin-bigons implies thin triangles (although the triangles can a lot thicker than the bigons).

I think this was proved by Papasoglou. Again, I don't have any reference materials with me, but this would be a nice thing to include. TheAstonishingBadger 23:52, 9 November 2007 (UTC)

[edit] Illustrations

This article's appeal will increase at least by a factor of ten if we can put in a good picture illustrating the delta-thin triangle property. Help! Arcfrk (talk) 14:50, 9 February 2008 (UTC)