Secondary cup and cap products in coarse geometry

We construct secondary cup and cap products on coarse (co-)homology theories from given cross and slant products. They are defined for coarse spaces relative to weak generalized controlled deformation retracts. On ordinary coarse cohomology, our secondary cup product agrees with a secondary product defined by Roe. For coarsifications of topological coarse (co-)homology theories, our secondary cup and cap products correspond to the primary cup and cap products on Higson dominated coronas via transgression maps. And in the case of coarse $$\mathrm {K}$$ K -theory and -homology, the secondary products correspond to canonical primary products between the $$\mathrm {K}$$ K -theories of the stable Higson corona and the Roe algebra under assembly and co-assembly.

Coarse quotients by group actions and the maximal Roe algebra

For a finitely generated group [Formula: see text] acting on a metric space [Formula: see text], Roe defined the warped space [Formula: see text], which one can view as a kind of large scale quotient of [Formula: see text] by the action of [Formula: see text]. In this paper, we generalize this notion to the setting of actions of arbitrary groups on large scale spaces. We then restrict our attention to what we call coarsely discontinuous actions by coarse equivalences and show that for such actions the group [Formula: see text] can be recovered as an appropriately defined automorphism group [Formula: see text] when [Formula: see text] satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group [Formula: see text] on a discrete bounded geometry metric space [Formula: see text] there is a relation between the maximal Roe algebras of [Formula: see text] and [Formula: see text], namely that there is a ∗-isomorphism [Formula: see text], where [Formula: see text] is the ideal of compact operators. If [Formula: see text] has Property A and [Formula: see text] is amenable, then [Formula: see text] has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

Coarse indices of twisted operators

Several formulas for computing coarse indices of twisted Dirac type operators are introduced. One type of such formulas is by composition product in [Formula: see text]-theory. The other type is by module multiplications in [Formula: see text]-theory, which also yields an index theoretic interpretation of the duality between Roe algebra and stable Higson corona.

Smooth subalgebras of the Roe algebra

In this paper, for a discrete group with property (RD), we construct a smooth subalgebra of a certain subalgebra in the (uniform) Roe algebra of this group. And using Lafforgue’s [Formula: see text]-Theory, under certain conditions, we prove that this certain subalgebra and the (uniform) Roe algebra have the same [Formula: see text]-theory groups. Moreover, our smooth subalgebra and the (uniform) Roe algebra have the same [Formula: see text]-theory groups. Our result can be viewed as a smooth subalgebra construction of the (uniform) Roe algebra, which is, as far as we know, the first occurrence in literature.

CHARACTERIZATIONS OF LOCALLY FINITE ACTIONS OF GROUPS ON SETS

We show that an action of a group on a set X is locally finite if and only if X is not equidecomposable with a proper subset of itself. As a consequence, a group is locally finite if and only if its uniform Roe algebra is finite.

Coronae of relatively hyperbolic groups and coarse cohomologies

We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the [Formula: see text]-homology of the corona with the [Formula: see text]-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the [Formula: see text]-theory of the corona with the [Formula: see text]-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the [Formula: see text]-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.