On the \(K\)-theory of groups with finite asymptotic dimension. (English) Zbl 1144.19001
J. Reine Angew. Math. 612, 35-57 (2007); erratum ibid. 726, 291-292 (2017).
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 \(\Gamma\).
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 \(\Gamma\).
Reviewer: Takahiro Sudo (Nishihara)
MSC:
19A31 | \(K_0\) of group rings and orders |
19D50 | Computations of higher \(K\)-theory of rings |
19B28 | \(K_1\) of group rings and orders |
19G24 | \(L\)-theory of group rings |
22D15 | Group algebras of locally compact groups |
22E40 | Discrete subgroups of Lie groups |
Citations:
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