Refine search
Actions for selected content:
1 results
ASYMPTOTIC DIMENSION AND GEOMETRIC DECOMPOSITIONS IN DIMENSIONS 3 AND 4
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 119 / Issue 2 / October 2025
- Published online by Cambridge University Press:
- 21 April 2025, pp. 176-201
- Print publication:
- October 2025
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
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.
