What is a superrigid subgroup?

This is an expository paper. It is well known that a linear transformation can be defined to have any desired action on a basis. From this fact, one can show that every group homomorphism from Z^k to R^d extends to a homomorphism from R^k to R^d, and we will see other examples of discrete subgroups H of connected groups G, such that the homomorphisms defined on $H$ can (“almost”) be extended to homomorphisms defined on all of G. This is related to a very classical topic in geometry, the study of linkages.