Question (Gromov). Classify groups up to quasi-isometry.

1) Ends. Roughly, if X is a metric space, Ends(X) is the number of components of the boundary at \infty of X. If X = (\Gamma, d_s), then Ends captures algebraic information.

Definition. For functions f_1, f_2 : \mathbb{N} \longrightarrow \mathbb{ N}, we say f_1 \preceq f_2 if there exists C such that f_1(n) \leq C f_2 (Cn + C) + Cn + C.  If f_1 \preceq f_2 and f_1 \succeq f_2 then f_1 \simeq f_2.

2) Growth. If \Gamma is a group and S is a finite generating set.

f_\Gamma (n) = \# B(1, n),

where B(1, n) is the set of elements \gamma \in \Gamma such that l_S(\gamma) \leq n. This is a quasi-isometric invariant of \Gamma.

Example. f_{\mathbb{Z}^k}(n) \simeq n^k

Example. f_{F_2}(n) is exponential.

3) If \Gamma is finitely presented and \Gamma' is quasi-isometric to \Gamma then \Gamma' is also finitely presented.

4) Let \Gamma = \langle S | R \rangle is finitely presented; so \Gamma = F_S / \langle \langle R \rangle \rangle. Let r \in \langle \langle R \rangle \rangle. Then

(1)     \prod_{i=1}^{n}g_{i}r_{i}^{{\varepsilon}_i}g_{i}^{-1}

where r_i \in R, \varepsilon_i \in \left\{ \pm 1 \right\}, and g_i \in \Gamma.  The question: how hard is it to write r in such a product? Define Area (r) to be minimum n in any such expression of r in (1). Let

\delta_\Gamma(n) = \max \left\{ Area(r) | r \in \langle \langle R \rangle \rangle, l_S(r) \leq n \right\}

This function \delta_\Gamma is the Dehn function of \Gamma, which measures how hard the word problem is to solve in \Gamma.  The \simeq-class of \delta_\Gamma 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, X is always a geodesic metric space.  We’ll write {[x,y]} for a geodesic between x and y (not necessarily unique).

Definition. Let x,y,z \in X, and let \bigtriangleup = [x,y] \cup [y,z] \cup [z,x].  We say that \bigtriangleup is \deltaslim if

{ [y,z] \subseteq B ( [x,y] \cup [z,x], \delta)},

where B(A, \delta) = \bigcup_{a \in A} B(a, \delta), and the same for both {[x,y]} and {[z,x]} (that is, for each geodesic “side” of the triangle, it is contained in a \delta neighborhood of the other two geodesic sides of the triangle).

Definition. X is Gromov hyperbolic (or \deltahyperbolic, or just hyperbolic) if every geodesic triangle, \bigtriangleup, is uniformly \delta-slim; that is, there exists \delta such that every \bigtriangleup is \delta-slim.

Example (a). Any tree is {0}-hyperbolic.  Every geodesic triangle is a “tripod”.

Example (b). \mathbb{R}^2 is not \delta-hyperbolic for any \delta.

Example (c). \mathbb{H}^2 (and hence \mathbb{H}^n) is hyperbolic (and indeed, any space of principal negative sectional curvature bounded away from zero).

Given a geodesic triangle \bigtriangleup \subseteq \mathbb{H}^2 and let a \in \bigtriangleup.  We ask how far from the other sides is a? Well, inscribe a semi-circle centered at a inside of \bigtriangleup; pick the largest such inscribed semi-circle, and call its radius \delta_a.  So \bigtriangleup is \delta-slim, where \delta is the largest \delta_a; that is, \delta is the radius of the largest semi-circle that can be inscribed in \bigtriangleup.

So to find \delta, we look at semi-circles; for this, we need a fact about \mathbb{H}^2.

Fact. For any \bigtriangleup \subseteq \mathbb{H}^2, Area(\bigtriangleup) = \pi - \alpha - \beta - \gamma < \pi, where \alpha, \beta, \gamma are  angles of the triangle.

This leads to a uniform bound on the area, and hence the radius of semi-circles inscribed in \bigtriangleup.

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 \delta-hyperbolic.

Definition. A quasi-geodesic is a quasi-isometric embedding of a closed interval.

Exercise 13. Let c : [1,\infty) \longrightarrow \mathbb{R}^2 by c(t) = (t, \log t) in polar coordinates. Show that c is a quasi-isometric embedding.

We will prove this behavior does not happen in hyperbolic metric spaces.