Nielsen realization for sphere twists on 3-manifolds

Abstract For a 3-manifold M , the twist group Twist( M ) is the subgroup of the mapping class group Mod( M ) generated by twists about embedded 2-spheres. We study the Nielsen realization problem for subgroups of Twist( M ). We prove that a nontrivial subgroup G < Twist( M ) is realized by diffeomorphisms if and only if G is cyclic and M is a connected sum of lens spaces, including S 1 × S 2 . We also apply our methods to the Burnside problem for 3-manifolds and show that Diff( M ) does not contain an infinite torsion group when M is reducible and not a connected sum of lens spaces.