  Given a non-compact, simply connected homogeneous three-manifold $X$ and a
sequence $\{\Omega_n\}_n$ of isoperimetric domains in $X$ with volumes tending
to infinity, we prove that as $n\to \infty $: 1. The radii of the $\Omega_n$
tend to infinity. 2. The ratios $\{Area} (\partial \Omega_n)/\{Vol}(\Omega_n)$
converge to the Cheeger constant Ch$(X)$, which we also prove to be equal to
$2H(X)$ where $H(X)$ is the critical mean curvature of $X$. 3. The values of
the constant mean curvatures $H_n$ of the boundary surfaces $\partial \Omega_n$
converge to $\frac{1}{2}\{Ch}(X)$. Furthermore, when Ch$(X)$ is positive, we
prove that for $n$ large, $\partial \Omega_n$ is well-approximated in a natural
sense by the leaves of a certain foliation of $X$, where every leaf of the
foliation is a surface of constant mean curvature $H(X)$.
