Superrigid Subgroups and Syndetic Hulls in Solvable Lie Groups

It is not difficult to see that every group homomorphism from Z to R extends to a homomorphism from R to R. We discuss other examples of discrete subgroups Γ of connected Lie groups G, such that the homomorphisms defined on Γ can (“virtually”) be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that Γ has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls. 1 What is a Superrigid Subgroup? Let us begin with a trivial example of the type of theorem that we will discuss. It follows easily from the fact that a linear transformation can be defined to have any desired action on a basis. (See Sect. 3 for a more complicated proof.) Proposition 1.1. Any group homomorphism φ:Z → R extends to a continuous homomorphism φ̂:R → R. A superrigidity theorem is a version of this simple proposition in the situation where Z, R, and R are replaced by more interesting groups: Suppose Γ is a discrete subgroup of a connected Lie group G, and H is some other Lie group. Does every homomorphism φ:Γ → H extend to a continuous homomorphism φ̂ defined on all of G? All of the Lie groups we consider are assumed to be linear groups ; that is, they are subgroups of GL(l,C), for some l. For example, R can be thought of as a linear group; in particular: R 3 ∼=   1 0 0 R 0 1 0 R 0 0 1 R 0 0 0 1   . (1) Thus, any homomorphism into R can be thought of as a homomorphism into GL(d + 1,R). The study of homomorphisms into GL(d,R) or GL(d,C) is known as Representation Theory. Unfortunately, in this much more interesting setting, not all homomorphisms extend. Proposition 1.2. There is a group homomorphism φ:Z → GL(d,R) that does not have a continuous extension to a homomorphism φ̂:R → GL(d,R).