Centralizers in Domains of Finite Gelfand-kirillov Dimension

We study centralizers of elements in domains. We generalize a result of the author and Small [4], showing that if A is a finitely generated noetherian domain and a ∈ A is not algebraic over the extended centre of A then the centralizer of a has Gelfand-Kirillov dimension at most one less than the Gelfand-Kirillov dimension of A. In the case that A is a finitely generated noetherian domain of GK dimension 3 over the complex numbers, we show that the centralizer of an element a ∈ A that is not algebraic over the extended centre of A satisfies a polynomial identity.