We prove that for a finitely generated linear group over a field of
positive characteristic the family of quotients by finite subgroups
has finite asymptotic dimension. We use this to show that the
-theoretic
assembly map for the family of finite subgroups is split injective for every finitely generated
linear group
over a commutative ring with unit under the assumption that
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
exists.
Keywords
$K$- and $L$-theory of group rings, injectivity of the
assembly map, linear groups
