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) $r \in \langle \langle R \rangle \rangle$

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 $\delta$slim 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 $\delta$hyperbolic, 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.