We show some surprisingly strong rigidity theorems for quasi-isometric embeddings of higher rank non-uniform lattices.

For example any QI embedding of SL(n, Z) into SL(n, Z[i]) is at bounded distance from a homomorphism on a subgroup of finite index.  This is the first example of a fairly general rigidity theorem but there are also examples that show some additional flexibility. A key step in our proofs is a variant of Ratner's theorem due to N.Shah. This is joint work with Thang Nguyen.