**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.**

**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

(1)

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.

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12 February 2009 at 9.26 pm

Henry WiltonThanks, Mark. I realized I’d used slightly stupid notation for the ball in the definition of growth, so I changed it.