A group with unbounded dead-end depth for all finite generating sets

The dead-end depth of a group element $g\in G$ with respect to a finite
generating set $A$ is defined as the distance (with respect to $A$) from
$g$ to the complement of the radius-$d_A(1,g)$ closed ball about the
identity in $G$ (where $d_A$ denotes distance with respect to $A$). We
show that the discrete Heisenberg group has elements (“dead ends”) of
arbitrarily large depth with respect to any finite generating set. This
is the first group known to have this property. We also show that these
dead ends have a curious property, namely that they can be connected to
the complement of the closed balls without going much closer to the
identity.