### Endomorphisms of semigroups: growth and interactions with subsemigroups (Alan Cain, NBSAN York, 20th November 2013)

Abstract: The growth of an endomorphism of a semigroup is (informally) a measure of how much a ball in the Cayley graph can be stretched by an endomorphism, taking the limit as the radius of the ball goes to infinity. I will introduce the concept and survey some of its properties. In particular, for any real number r \geq 1, there is an endomorphism whose growth is r. On the other hand, the growth rate of an endomorphism of a semigroup with a homogeneous presentation must be an algebraic number. I will also describe how the growth of an endomorphism can be connected (or not connected) to subsemigroups that it respects.(This is joint work with Victor Maltcev.)