On the transfer reducibility of certain Farrell-Hsiang groups

We show how the existing proof of the Farrell-Jones Conjecture for virtually poly-$\mathbb{Z}$-groups can be improved to rely only on the usual inheritance properties in combination with transfer reducibility as a sufficient criterion for the validity of the conjecture.Comment: 18 page

Algebraic K-theory of stable ∞\infty-categories via binary complexes

We adapt Grayson's model of higher algebraic $K$-theory using binary acyclic complexes to the setting of stable $\infty$-categories. As an application, we prove that the $K$-theory of stable $\infty$-categories preserves infinite products.Comment: 20 pages; accepted for publication by the Journal of Topolog

K1K_1-groups via binary complexes of fixed length

We modify Grayson's model of $K_1$ of an exact category to give a presentation whose generators are binary acyclic complexes of length at most $k$ for any given $k \ge 2$. As a corollary, we obtain another, very short proof of the identification of Nenashev's and Grayson's presentations.Comment: 10 pages, minor changes following a referee report, to appear in HH

The A-theoretic Farrell–Jones conjecture for virtually solvable groups

We prove the A -theoretic Farrell–Jones conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for S -arithmetic groups and lattices in almost connected Lie groups