  We show that the fundamental groups of smooth $4$-manifolds that admit
geometric decompositions in the sense of Thurston have asymptotic dimension at
most four, and equal to 4 when aspherical. We also show that closed
$3$-manifold groups have asymptotic dimension at most 3. Our proof method
yields that the asymptotic dimension of closed $3$-dimensional Alexandrov
spaces is at most 3. We thus obtain that the Novikov conjecture holds for
closed $4$-manifolds with such a geometric decomposition and closed
$3$-dimensional Alexandrov spaces. Consequences of these results include a
vanishing result for the Yamabe invariant of certain $0$-surgered geometric
$4$-manifolds and the existence of zero in the spectrum of aspherical smooth
$4$-manifolds with a geometric decomposition.
