On the K‐theory of groups with finite decomposition complexity

It is proved that the assembly map in algebraic K‐ and L‐theory with respect to the family of finite subgroups is injective for groups Γ with finite quotient finite decomposition complexity (a strengthening of finite decomposition complexity introduced by Guentner, Tessera and Yu) that admit a finite‐dimensional model for E̲Γ and have an upper bound on the order of their finite subgroups. In particular, this applies to finitely generated linear groups over fields with characteristic zero with a finite‐dimensional model for E̲Γ.