  We prove that for a finitely generated linear group G over a field of
positive characteristic the family of quotients by finite subgroups has finite
asymptotic dimension. We use this to show that the K-theoretic assembly map for
the family of finite subgroups is split injective for every finitely generated
linear group G over a commutative ring with unit under the assumption that G
admits a finite-dimensional model for the classifying space for the family of
finite subgroups. Furthermore, we prove that this is the case if and only if an
upper bound on the rank of the solvable subgroups of G exists.
