Extending the results from A. C. Bartels [\(K\)-Theory 28, No. 1, 19–37 (2003; Zbl 1036.19002)] by relaxing the finite \(B\Gamma\) assumption for a discrete group \(\Gamma\), it is proved that the assembly map in algebraic \(K\)-theory is a split injection, where it is assumed that there is a finite-dimensional \(\Gamma\)-CW-model for the universal space for proper \(\Gamma\)-actions, and there is a \(\Gamma\)-invariant metric on the space such that the space is uniformly contractible in some sense, and is a complete proper path metric space, and has finite asymptotic dimension.
Consequently, it follows that the assembly map in algebraic \(K\)-theory is split injective for \(\Gamma\) such that either it is a discrete subgroup of a virtually connected Lie group, or it has finite asymptotic dimension and admits a cocompact \(\Gamma\)-CW-model for the universal space. Furthermore, it is deduced that the assemply map in \(L\)-theory is a split injection under the same assumption and an additional one on vanishing of (negative) higher \(K\)-groups of the group rings for finite subgroups of \(\Gamma\).