Endomorphisms of semigroups: growth and interactions with subsemigroups

This talk is divided into two parts: the first will cover the concept of the growth of an endomorphism of a subsemigroup. Informally, this is a measure of how much balls in the Cayley graph of a semigroup are stretched by iterations of the endomorphism. I will describe how every real number $r > 1$ arises as the growth of some semigroup endomorphism, but the main focus will be on how the growth of an endomorphism of a semigroup interacts with the growth of its restriction to suitable subsemigroups (which must be preserved by the endomorphism). The second part of the talk examines hopficity and co-hopficity. A semigroup is hopfian if every surjective endomorphism is also injective (equivalently, the semigroup is not a proper homomorphic image of itself). A semigroup is co-hopfian if every injective endomorphism is also surjective (equivalently, the semigroup is not isomorphic to any proper subsemigroup of itself). Maltcev and Ruškuc proved that if $S$ is a finite Rees index extension of a finitely generated hopfian semigroup $T$, then $S$ is itself hopfian, but that the converse does not hold, and that the result does not hold without the hypothesis of finite generation. I will describe the corresponding results for co-hopficity. This is joint work with Victor Maltcev.