  We make use of $\mathcal{F}$-structures and technology developed by Paternain
- Petean to compute minimal entropy, minimal volume, and Yamabe invariant of
symplectic 4-manifolds, as well as to study their collapse with sectional
curvature bounded from below. \`A la Gompf, we show that these invariants
vanish on symplectic 4-manifolds that realize any given finitely presented
group as their fundamental group. We extend to the symplectic realm a result of
LeBrun which relates the Kodaira dimension with the Yamabe invariant of compact
complex surfaces.
