#### Vol. 1, No. 4, 2016

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Splitting the relative assembly map, Nil-terms and involutions

### Wolfgang Lück and Wolfgang Steimle

Vol. 1 (2016), No. 4, 339–377
##### Abstract

We show that the relative Farrell–Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic $K\phantom{\rule{0.3em}{0ex}}$-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 $ℤ\left[ℤ∕2\right]$-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.

##### Keywords
splitting relative $K$-theoretic assembly maps, rational vanishing and Tate cohomology of the relative Nil-term
##### Mathematical Subject Classification 2010
Primary: 18F25, 19A31, 19B28, 19D35