Milnor-Wolf Theorem for group endomorphisms

We study the growth of group endomorphisms and we prove an analogue of Chou’s extension of Milnor-Wolf Theorem. Indeed, if G is an elementary amenable group and φ:G→G is an endomorphism, then φ has either polynomial or exponential growth. This result follows by studying the growth of automorphisms of finitely generated groups, where we prove some stronger results.