Generating pairs and group actions

An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G. We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group A(5), and give various other examples. (C) 2013 Elsevier B.V. All rights reserved.