In his book Topics in Geometric Group Theory, Pierre de la Harpe calls the following result the Fundamental Observation of Geometric Group Theory (though he also calls it a theorem!). It is also often called the Svarc--Milnor Lemma. Roughly speaking, it asserts that the coarse geometry of a group is captured by any suitably nice action of that group by isometries on a metric space.

Theorem. Let $X$ be a metric space that is geodesic and proper, let $\Gamma$ be a group and let $\Gamma$ act properly discontinuously and cocompactly by isometries on $X$. Then $\Gamma$ is finitely generated, and furthermore for any $x_0\in X$ the map $\Gamma\to X$ given by

$\gamma\mapsto\gamma x_0$

is a quasi-isometry.

Remarks.

1. $\Gamma$ is endowed with the word metric (with respect to some choice of finite generating set).
2. A map of metric space $f:Y\to X$ is a quasi-isometric embedding if there are constants $\lambda\geq 1$, $\mu\geq 0$ such that

$\lambda d_Y(y_1,y_2)+\mu\geq d_X(f(y_1),f(y_2))\geq \frac{1}{\lambda} d_Y(y_1,y_2)-\mu$

for all $y_1,y_2\in Y$. It is a quasi-isometry if, furthermore, for every $x\in X$ there is $y\in Y$ such that $d(x,f(y))\leq \mu$.