# Members & Guests

## Project

10Duality and the coarse assembly map

## Publications within SPP2026

We strengthen a result of Hanke-Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group C*-algebra even holds for the reduced group C*-algebra. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

Related project(s):
10Duality and the coarse assembly map

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.

Related project(s):
10Duality and the coarse assembly map

We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic K-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to prove injectivity results about assembly maps.

Related project(s):
10Duality and the coarse assembly map

This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible $$\sigma$$-compact space in which the corona sits as a $$\mathbb{Z}$$-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space $$BG$$, then our constructions yield a $$\mathbb{Z}$$-structure for the group.