Representation stability asks whether an approximate representation of a group can be approximated by an actual representation.  There are many technical variations of this basic question: I will focus mainly on approximate representations into finite-dimensional unitary groups.

I’ll introduce the two properties in the title - the LLP of Kirchberg and property FD of Lubotzky-Shalom - via group C*-algebras and explain how they imply some fairly weak representation stability results.  I’ll then explain some refinements you can get using K-theory (without assuming any background knowledge of K-theory).  Finally, I'll discuss the known range of validity of the LLP and property FD (I’ll also mention some related properties like Kechris’ property MD).

The non K-theoretic parts are based on joint work with Francesco Fournier-Facio.