We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the category of equivariant coarse motivic spectra.As examples of equivariant coarse homology theories we discuss equivariant coarse ordinary homology and equivariant coarse algebraic K-homology.Moreover, we discuss the cone functor, its relation with equivariant homology theories in equivariant topology, and assembly and forget-control maps. This is a preparation for applications in subsequent papers aiming at split-injectivity results for the Farrell-Jones assembly map.

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We prove the Farrell-Jones Conjecture for (non-connective) Atheory with coefficients and finite wreath products for hyperbolic groups, CAT(0)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudo-isotopies in the topological, piecewise linear and smooth category.2010 Mathematics Subject Classification. 19D10, 57Q10, 57Q60.

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 generally, groups with finite decomposition complexity. Contents

Using the language of coarse homology theories, we provide an axiomatic account of vanishing results for the fibres of forget-control maps associated to spaces with equivariant finite decomposition complexity. *