Let C,D be classes of finitely presented groups. The epimorphism problem from C to D is the following decision problem:
Input: Finite descriptions (presentation, multiplication table, other) for groups G in C and H in D
Question: Is there an epimorphism from G to H?
I will discuss some cases where it is decidable and where it is NP-complete. Spoiler alert: it is undecidable for C=D=the class of 2-step nilpotent groups (Remeslennikov).
This is joint work with Jerry Shen (UTS) and Armin Weiss (Stuttgart).