The Farrell–Jones conjecture for normally poly-free groups

Abstract

We construct explicit finite-dimensional orthogonal representations πN of SLN (Z) for N ∈ {3, 4} all of whose invariant vectors are trivial, and such that H N −1 (SLN (Z), πN ) is non-trivial. This implies that for N as above, the group SLN (Z) does not have property (TN−1) of Bader-Sauer and therefore is not (N − 1)-Kazhdan in the sense of De Chiffre-Glebsky-Lubotzky-Thom, both being higher versions of Kazhdan's property T .

The Farrell-Jones Conjecture for hyperbolic-by-cyclic groups

“…Theorem 1.1 generalizes the case when G is a finite rank free group, which we proved in [BFW], and the present paper supersedes it, so [BFW] will not be published independently. Brück, Kielak and Wu generalized this result to infinitely generated free groups and further to normally poly-free groups in [BKW21].…”