The main goal of the thesis is to construct equivariant coarse versions of the classical Hochschild and cyclic homologies of algebras. These are lax symmetric monoidal functors from the category of equivariant bornological coarse spaces to the cocomplete stable ∞-category of chain complexes and are called equivariant coarse Hochschild and cyclic homology. If k is a field, the evaluation at the one point bornological coarse space induces equivalences with the classical Hochschild and cyclic homologies of k. In the equivariant setting, the G-equivariant coarse Hochschild (cyclic) homology of (a canonical bornological coarse space associated to) the group G agrees with the classical Hochschild
(cyclic) homology of the associated group algebra k[G].
The second aim of the thesis is the construction of natural transformations from
equivariant coarse algebraic K-homology to equivariant coarse Hochschild and cyclic homology, and of natural transformations from equivariant coarse Hochschild and cyclic homology to equivariant coarse ordinary homology. This is achieved by using trace-like maps and gives a natural transformation from equivariant coarse algebraic K-homology to equivariant coarse ordinary homology.
We conclude the dissertation with two additional investigations: we give a comparison result between the forget-control map for equivariant coarse Hochschild homology and the associated generalized assembly map, and we show a Segal-type localization theorem for equivariant Hochschild and cyclic homology.