We show that the relative Farrell–Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic $K$-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a $\mathbb{Z}\left[\mathbb{Z}∕2\right]$-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.

We define a category of pure birational motives over a field, depending on the choice of an adequate equivalence relation on algebraic cycles. It is obtained by “killing” the Lefschetz motive in the corresponding category of effective motives. For rational equivalence, it encompasses Bloch’s decomposition of the diagonal. We study the induced Chow–Künneth decompositions in this category, and establish relationships with Rost’s cycle modules and the Albanese functor for smooth projective varieties.

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 $K$-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.

We prove that the standard norm varieties for Milnor symbols mod $p$ of length $n$ are birationally isomorphic to Pfister quadrics when $p=2$, to Severi–Brauer varieties when $p>2$ and $n=2$, and to varieties defined by reduced norms of cyclic algebras when $p>2$ and $n=3$. In the case $p=2$ and the case $p>2$ and $n=2$, the results imply that the standard norm varieties for two equal Milnor symbols mod $p$ are birationally isomorphic, and we conjecture this in general.