Talk:Gromov boundary

Ends?
Any explanation how this concept relates to or is distinct from the notions of end (topology) and end (graph theory)? —David Eppstein (talk) 03:32, 28 November 2013 (UTC)


 * I've added those in. Brirush (talk) 17:37, 28 November 2013 (UTC)

Hyperbolic spaces are tree-like?
"Since hyperbolic spaces are tree-like..." What is that supposed to mean?--79.205.95.29 (talk) 15:50, 5 September 2017 (UTC)


 * Roughly, this means that at large scales finite sets of points have the same metric properties as trees. See https://en.wikipedia.org/wiki/Hyperbolic_metric_space#Approximate_trees_in_hyperbolic_spaces or https://en.wikipedia.org/wiki/Hyperbolic_metric_space#Asymptotic_cones for rigorous statements about this. jraimbau (talk) 12:27, 6 September 2017 (UTC)

Can this be right for all compact Riemann surfaces?
One sentence states:

"The Gromov boundary of the fundamental group of a compact Riemann surface is the unit circle."

Somehow I doubt this can be right for the sphere S2.

Is it correct for the torus T2? Or is it true only for surfaces of genus ≥ 2 ?

I hope someone knowledgeable about this subject can fix this.


 * These two are special cases which are excluded from the general theory referred to in this sentence. The boundary of the sphere is empty ; the fundamental group of the torus is not hyperbolic so the definition does not apply. I added the stipulation that the surface be hyperbolic to this part of the article. jraimbau (talk) 08:01, 3 February 2024 (UTC)

Definition of the boundary of a group is never given. Yet it is used.
The section Properties of the Gromov boundary mentions one situation in which it is applied to groups.

But no general definition of the Gromov boundary of a group is in this article.

Nevertheless, in the section Examples there are two examples of what is described as the Gromov boundary of a group.

I hope someone knowledgeable about this subject will include in the article a general definition of the Gromov boundary of a group.

(Or at least a statement of which groups this applies to and a definition of the Gromov boundary of such groups.)


 * I added the definition in this section and changed its name to "Gromov boundary of a group". jraimbau (talk)