  We work in a first-order setting where structures are spread out over a
metric space, with quantification allowed only over bounded subsets. Assuming a
doubling property for the metric space, we define a canonical {\em core}
$\mathcal{J}$ associated to such a theory, a locally compact structure that
embeds into the type space over any model. The automorphism group of
$\mathcal{J}$, modulo certain infinitesimal automorphisms, is a locally compact
group $\mathcal{G}$. The automorphism groups of models of the theory are
related with $\mathcal{G}$, not in general via a homomorphism, but by a {\em
quasi-homomorphism}, respecting multiplication up to a certain canonical
compact error set. This fundamental structure is applied to describe the nature
of approximate subgroups. Specifically we obtain a full classification of
(properly) approximate lattices of $SL_n({\mathbb{R}})$ or
$SL_n({\mathbb{Q}}_p)$.
