The main goals of our research program are the construction of a

geometric homological equivariant Chern character for proper actions of a

totally disconnected locally compact group $G$. The main building block for

this construction will be the new cycle model of $K^G_*(X)$ which is another

major goal of this research project. The cycles will be of the form $(M,E,f)$,

and the main features will be that there is a suitable version of

K-orientation encoded in $(M,E)$. Here $E$, which is of K-theoretic character,

will involve maps to a convenient classifying space for K-theory with a good

Chern character cocycle. All together, this should allow to construct a Chern

character cycle out of this data for a cosheaf on $G\backslash M$ encoding the

representation rings of the isotropy subgroup (with cosheaf structure given by

induction of representations), then to be pushed to the correct target depending on $X$.

The program will need quite a few innovations to accommodate non-compact

non-discrete totally disconnected groups. But it will follow in suitable parts

the constructions for the profinite situation of \cite{BaumSchneider}, as well

as the constructions for the discrete case described with related methods

in \cite{Raven}.

Indeed, the constructions should be sufficiently flexible to combine with the

K-theoretic Chern character we plan to develop and to provide a geometrically

defined equivariant \emph{bivariant} Chern character for actions of totally

disconnected groups.

We envision applications of such a homological Chern character toward

equivariant index theory of a new kind for actions of p-adic groups.

## Publications

We prove the following Lipschitz rigidity result in scalar curvature geometry. Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by $n(n-1)$, and let $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of non-zero degree to the unit $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his "four lectures". Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles, and on the theory of quasiregular maps due to Reshetnyak.

**Related project(s):****42**Spin obstructions to metrics of positive scalar curvature on nonspin manifolds**52**Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II**58**Profinite perspectives on l2-cohomology**73**Geometric Chern characters in p-adic equivariant K-theory

## Team Members

**Dr. Thorben Kastenholz**

Universität Göttingen

thorben.kastenholz(at)mathematik.uni-goettingen.de

**Prof. Dr. Thomas Schick**

Project leader

Georg-August-Universität Göttingen

thomas.schick(at)math.uni-goettingen.de