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**Lemma 27** *Revisited*. Suppose is a covering map. Then there is a covering map such that is the fibre product of and .

**Proof.** Let be the fibre product of and . There is a map given by . Let be the fibre product of and ; i.e.

.

There is a map given by . This is a covering map and injective, so it is a homeomorphism.

Let be continuous, be a covering map and , choices of basepoint. We have already seen that a choice of such that determines an elevation of to at . Fix such a . The pre-image of in is in bijection with the set of cosets

This raises the question, when do two cosets determine the same elevation?

**Exercise 24**. and determine the same elevation if and only if

;

that is, the set of elevations of to is in bijection with .

Let and be graphs of spaces and suppose we have the following data.

(a) A combinatorial map given by and .

(b) Covering maps for each .

(c) Covering maps for each , such that whenever adjoins .

This determines a continuous map . When is really a covering map?

**Theorem 17. ** is a covering map if

(i) for all adjoining , the edge map is an elevation of ; and

(ii) wherever adjoining and , every elevation to arises as an edge map of .

**Proof** (sketch). It’s enough to consider our local model : and . and be covering maps defining and a map . By Lemma 27, is a covering map if and only if is a fibre product with respect to some covering:

Every map in the diagram is injective, so (for each component)

and it follows that is the fibre product of and . The result follows.

**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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