Subfields of division algebras

Let A be a finitely generated domain of GK dimension less than 3 over a field K and let Q(A) denote the quotient division algebra of A. Using the ideas of Smoktunowicz, we show that if D is a finitely generated division subalgebras of Q(A) of GK dimension at least 2, then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dimension less than 3 then its subfields have transcendence degree at most 1 and, in particular, over an algebraically closed field any division subalgebra D of Q(A) is either commutative or has the property that Q(A) is finite dimensional as a left and right D-vector space. Finally, we study subfields of quotient division algebras of domains of finite GK dimension and introduce a combinatorial property we call the straightening property. We show that many classes of algebras have this straightening property and we show that if A is a domain of GK dimension d with the straightening property that is not PI, then the maximal subfields of Q(A) have transcendence degree at most d − 1, proving a special case of a conjecture of Small.