  Let k be an algebraically closed field, let K/k be a finitely generated field
extension of transcendence degree 2 with automorphism sigma, and let A be an
N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for
all n. Then if A is big enough in Q in an appropriate sense, we prove that
GKdim A = 3,4,5 or is infinite, with the exact value depending only on the
geometric properties of sigma. The proof uses techniques in the birational
geometry of surfaces which are of independent interest.
