On symplectic 4-manifolds with prescribed fundamental group

In this article we study the problem of minimizing $a\chi+b\sigma$ on the class of all symplectic 4--manifolds with prescribed fundamental group $G$ ($\chi$ is the Euler characteristic, $\sigma$ is the signature, and $a,b\in \BR$), focusing on the important cases $\chi$, $\chi+\sigma$ and $2\chi+3\sigma$. In certain situations we can derive lower bounds for these functions and describe symplectic 4-manifolds which are minimizers. We derive an upper bound for the minimum of $\chi$ and $\chi+\sigma$ in terms of the presentation of $G$.