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 $\Gamma$ with finite quotient finite decomposition complexity (a strengthening of finite decomposition complexity introduced by Guentner, Tesser and Yu) that admit a finite dimensional model for $\underbar E\Gamma$ 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 $\underbar E\Gamma$.