Geometric automorphism groups of symplectic 4-manifolds

Abstract

Let $M$ be a closed, oriented, smooth $4-$manifold with intersection form $\Gamma$, $A(\Gamma)$ the automorphism group of $\Gamma$ and $D(M)$ the subgroup induced by orientation-preserving diffeomorphisms of $M$. In this note we study the question when $D(M)$ is of infinite index in $A(\Gamma)$ for a symplectic 4-manifold.