Asymptotic dimension and geometric decompositions in dimensions 3 and 4

Abstract

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.

arXiv comment22 pages, 2 images

References in corpus (12)

• On asymptotic dimension of groups
• Embedding relatively hyperbolic groups in products of trees
• Minimal entropy and geometric decompositions in dimension four
• No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds
• Rigidity of high dimensional graph manifolds
• The Zero-in-the-Spectrum Question
• Bounded geometry in relatively hyperbolic groups
• A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds
• The Novikov Conjecture
• $S^2$-bundles over 2-orbifolds
• Topological rigidity of higher graph manifolds
• Collapsing and group growth as obstructions to Einstein metrics on some smooth 4-manifolds