  We say A is a quasi-normal subgroup of the group G if the commensurator of A
in G is all of G. We develop geometric versions of commensurators in finitely
generated groups. In particular, g is an element of the commensurator of A in G
iff the Hausdorff distance between A and gA is finite.
  We show that a quasi-normal subgroup of a group is the kernel of a certain
map, and a subgroup of a finitely generated group is quasi-normal iff the
natural coset graph is locally finite. This last equivalence is particularly
useful for deriving asymptotic results for finitely generated groups. Our
primary goal in this paper is to develop the basic theory of quasi-normal
subgroups, comparing analogous results for normal subgroups and isolating
differences between quasi-normal and normal subgroups.
