Symplectomorphism Groups and Almost Complex Structures

This paper studies groups of symplectomorphisms of ruled surfaces M for symplectic forms with varying cohomology class. This cohomology class is characterised by the ratio μ of the size of the base to that of the fiber. By considering appropriate spaces of almost complex structures, we investigate how the topological type of these groups changes as μ increases. If the base is a sphere, this changes precisely when μ passes an integer, and, for general bases, it stabilizes as μ → ∞. Our results extend and make more precise some of the conclusions of Abreu–McDuff concerning the rational homotopy type of these groups for rational ruled surfaces.