Call a complex polynomial f(x,y) _expanding_ if there is e>0 such that for all sufficiently large finite sets A and B of complex numbers with |B| >= |A|, we have |f(A,B)| > |A|^{1+e}. A result of Elekes and Rónyai shows that the only non-expanding polynomials f(x,y) are those obtained from addition or multiplication by composing with unary polynomials. Thinking of B as parametrising a family of unary polynomials f_b(x) = f(x,b), we can see this conclusion as placing B in an algebraic group acting via f. Generalising in these terms, arbitrary nilpotent algebraic groups and their actions can arise. I will review some results indicating that this should be the most general situation, including work of Tingxiang Zou and myself which confirms this in certain cases, using methods from model theory, additive and incidence combinatorics, group theory, and number theory.