Abstract
We show injectivity results for assembly maps using equivariant coarse homology theories with transfers. Our method is based on the descent principle and applies to a large class of linear groups or, more general, groups with finite decomposition complexity.
Highlights
In [BEKW19, Def. 3.14] we defined the notion of G-equivariant finite decomposition complexity (G-FDC) for a G-coarse space (Definition 3.6)
We introduce the notion of an equivariant coarse homology theory; see [BEKW20, Sec. 3] for details
In order to capture continuity of equivariant coarse homology theories motivically we introduce the universal continuous equivariant coarse homology theory
Summary
For a group G we consider a functor M : GOrb → C from the orbit category of G to a cocomplete ∞-category C. In [BEKW19, Def. 3.14] we defined the notion of G-equivariant finite decomposition complexity (G-FDC) for a G-coarse space (Definition 3.6). By the transitivity principle [BL06, Thm. 2.4] the assembly map AsmblP∩FDC,KAG is an equivalence (here we have to use the assumptions on the groups Pi as well as that the Farrell–Jones conjecture passes to subgroups [BR07, Thm. 4.5]). Apply the construction of [Win[15], Def. 2.2] to obtain a G-simplicial complex X[Y, υ0] whose dimension is bounded by nk + n + k After observing that this construction is compatible with taking fixed points in the sense that X[Y, υ0]H ∼= XH[YH, υ0H] for all subgroups H of G, [Win[15], Cor. 2.5] implies that X[Y, υ0] is a model for EFtopG. Taking some Pi to be CAT(0)-groups that are not known to have FDC and some Pi to be groups that have FDC but for which the Farrell–Jones conjecture is not known, we obtain examples of groups for which Theorem 1.15 applies and the split-injectivity was not known before
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