A quasi-isometry between metric spaces 
and 
is a map 
for which there are constants 
and 
satisfying 
for
all 
,
and every point of 
is at distance at most 
from 
. The second condition is coarse surjectivity. A quasi-isometry
need not be a bijection or preserve small distances.
Word metrics from different finite generating sets of the same group are quasi-isometric.
A group is virtually polycyclic when it has a polycyclic group as a subgroup of finite index.
