We prove that abelian subgroups of the outer automorphism group of a free group are quasiisometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for Out(Fn).

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We prove the Farrell–Jones conjecture (FJC) for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the FJC by Bartels–Lück–Reich, as well as the structure theory of mapping tori by Dahmani–Krishna.

Suppose that M is a finitely-generated graded module (generated in degree 0) of codimension c ≥ 3 over a polynomial ring and that the regularity of M is at most 2a − 2 where a ≥ 2 is the minimal degree of a first syzygy of M . Then we show that the sum of the betti numbers of M is at least β0(M )(2 c + 2 c−1 ). Additionally, under the same hypothesis on the regularity, we establish the surprising fact that if c ≥ 9 then the first half of the betti numbers are each at least twice the bound predicted by the Buchsbaum-Eisenbud-Horrocks rank conjecture: for 1 ≤ i ≤ c+1 2 , βi(M ) ≥ 2β0(M ) c i .