# Geometric automorphism groups of symplectic 4-manifolds

## TL;DR

This paper investigates the relationship between the automorphism group of the intersection form of a symplectic 4-manifold and the subgroup induced by orientation-preserving diffeomorphisms, focusing on when the latter has infinite index.

## Contribution

It provides new insights into the conditions under which the diffeomorphism-induced subgroup has infinite index in the automorphism group for symplectic 4-manifolds.

## Key findings

- Identifies conditions for infinite index of D(M) in A(Γ)
- Analyzes the structure of automorphism groups in symplectic 4-manifolds
- Contributes to understanding symmetries of 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.