Jordan property of birational automorphism groups of surfaces and birational permutations

Abstract: We prove that the group of birational automorphisms of a geometrically irreducible algebraic surface over a finite field is Jordan. We show that the analogous statement fails in higher dimensions. Finally, we prove that groups of birational permutations over finite fields have bounded finite $p'$-subgroups; in particular, they are $p$-Jordan.