  We study the class of densely related groups. These are finitely generated
(or more generally, compactly generated locally compact) groups satisfying a
strong negation of being finitely presented, in the sense that new relations
appear at all scales. Here, new relations means relations that do not follow
from relations of smaller size. Being densely related is a quasi-isometry
invariant among finitely generated groups.
  We check that a densely related group has none of its asymptotic cones simply
connected. In particular a lacunary hyperbolic group cannot be densely related.
  We prove that the Grigorchuk group is densely related. We also show that a
finitely generated group that is (infinite locally finite)-by-cyclic and which
satisfies a law must be densely related. Given a class $\mathcal{C}$ of
finitely generated groups, we consider the following dichotomy: every group in
$\mathcal{C}$ is either finitely presented or densely related. We show that
this holds within the class of nilpotent-by-cyclic groups and the class of
metabelian groups. In contrast, this dichotomy is no longer true for the class
of $3$-step solvable groups.
