Actions on lens spaces which respect a Heegaard decomposition

Let L=L(p,q) be a 3-dimensional lens space. We consider smooth finite group actions on L which leave a Heegaard torus and each of its complementary components invariant. Such an action is said to have rotational type if each element is isotopic to the identity on L, and to have dirotational type otherwise. In this paper we enumerate and classify these two types of actions up to equivalence, where two actions are equivalent if their images are conjugate in the group of self-diffeomorphisms of L. When q2≠±1 (mod p) this results in a conjecturally complete classification of all finite group actions on L.