Any finitely presented group can be the fundamental of a smooth 4-manifold. So, generally, the study of 4-manifolds has been restricted to simply connected cases or, thanks to Freedman and Teichner's profound results, to groups with subexponential growth (where surgery theoretical results still hold). In this talk, we'll explore this a bit further. We will use two particular properties of groups to see how they help us understand smooth four-manifolds.

First, using the growth type in combination with sequences of volume-collapsing Riemannian metrics, we can rule out the existence of Einstein metrics.

Second, by computing the asymptotic dimension for manifolds with a geometric decomposition, we obtain a proof of Novikov's conjecture for that particular family of smooth 4-manifolds. As a consequence, we find a vanishing result for the Yamabe invariant of certain 0-surgery geometric 4-manifolds and the existence of zero in the spectrum of aspherical smooth 4-manifolds with a geometric decomposition. Moreover, our proof method also shows that closed 3-manifold groups and closed 3-dimensional Alexandrov spaces have asymptotic dimensions at most 3 (and exactly 3 when aspherical).

This is all joint work with Haydeé Contreras Peruyero.

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