ASYMPTOTIC DIMENSION AND GEOMETRIC DECOMPOSITIONS IN DIMENSIONS 3 AND 4

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 four when aspherical. We also show that closed $3$ -manifold groups have asymptotic dimension at most three. Our proof method yields that the asymptotic dimension of closed $3$ -dimensional Alexandrov spaces is at most three. Thus, we obtain that the Novikov conjecture holds for closed $4$ -manifolds with such a geometric decomposition and for 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.