Symplectic 4-manifolds with arbitrary fundamental group near the   Bogomolov-Miyaoka-Yau line
  In this paper we construct a family of symplectic 4--manifolds with positive signature for any given fundamental group $G$ that approaches the BMY line. The family is used to show that one cannot hope to do better than than the BMY inequality in finding a lower bound for the function $f=\chi+b\sigma$ on the class of all minimal symplectic 4-manifolds with a given fundamental group. 
Scott Baldridge, Paul Kirk