Gk-dimension of Birationally Commutative Surfaces

Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism σ ∈ Autk(K), and let A ⊆ Q = K[t;σ] be an N-graded subalgebra with dimk An < ∞ for all n ≥ 0. Then if A is big enough in Q in an appropriate sense, we prove that GKA = 3, 4, 5, or ∞, with the exact value depending only on the geometric properties of σ. The proof uses techniques in the birational geometry of surfaces which are of independent interest.