Question (Gromov). Classify groups up to quasi-isometry.
1) Ends. Roughly, if is a metric space, is the number of components of the boundary at of . If , then captures algebraic information.
Definition. For functions , , we say if there exists such that . If and then .
2) Growth. If is a group and is a finite generating set.
where is the set of elements such that . This is a quasi-isometric invariant of .
Example. is exponential.
3) If is finitely presented and is quasi-isometric to then is also finitely presented.
4) Let is finitely presented; so . Let . Then
where , , and . The question: how hard is it to write in such a product? Define to be minimum in any such expression of in (1). Let
This function is the Dehn function of , which measures how hard the word problem is to solve in . The -class of is a quasi-isometric invariant.
Remark. Having a solvable word problem is equivalent to having a computable Dehn function.
Hyperbolic Metric Spaces
We want a notion of metric spaces (and hence for groups) that captures hyperbolicity (that is, for one, that triangles are thin).
In what follows, is always a geodesic metric space. We’ll write for a geodesic between and (not necessarily unique).
Definition. Let , and let . We say that is -slim if
where , and the same for both and (that is, for each geodesic “side” of the triangle, it is contained in a neighborhood of the other two geodesic sides of the triangle).
Definition. is Gromov hyperbolic (or -hyperbolic, or just hyperbolic) if every geodesic triangle, , is uniformly -slim; that is, there exists such that every is -slim.
Example (a). Any tree is -hyperbolic. Every geodesic triangle is a “tripod”.
Example (b). is not -hyperbolic for any .
Example (c). (and hence ) is hyperbolic (and indeed, any space of principal negative sectional curvature bounded away from zero).
Given a geodesic triangle and let . We ask how far from the other sides is ? Well, inscribe a semi-circle centered at inside of ; pick the largest such inscribed semi-circle, and call its radius . So is -slim, where is the largest ; that is, is the radius of the largest semi-circle that can be inscribed in .
So to find , we look at semi-circles; for this, we need a fact about .
Fact. For any , , where are angles of the triangle.
This leads to a uniform bound on the area, and hence the radius of semi-circles inscribed in .
To define hyperbolic groups, we want to prove hyperbolicity is a quasi-isometric invariant of geodesic metric spaces. We need to “quasi-fy” the definition of -hyperbolic.
Definition. A quasi-geodesic is a quasi-isometric embedding of a closed interval.
Exercise 13. Let by in polar coordinates. Show that is a quasi-isometric embedding.
We will prove this behavior does not happen in hyperbolic metric spaces.