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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 forE  ̲   Γ    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 forE  ̲   Γ   .

On the K ‐theory of groups with finite decomposition complexity