The equivariant coarse Novikov conjecture and coarse embedding

Abstract

The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group $\Gamma$ acts on a bounded geometric space $X$ properly, isometrically, and with bounded distortion, $X/\Gamma$ and $\Gamma$ admit coarse embeddings into Hilbert space, then the $\Gamma$-equivariant coarse Novikov conjecture holds for $X$. Here bounded distortion means that for any $\gamma \in \Gamma$, ${\sup }_{x\in Y}d(\gamma x,x)<\infty$, where $Y$ is a fundamental domain of the $\Gamma$-action on $X$.

References in corpus (6)

• Coronae of product spaces and the Coarse Baum-Connes conjecture
• The coarse Baum-Connes conjecture for relatively hyperbolic groups
• Dynamical complexity and controlled operator K-theory
• The coarse Baum-Connes conjecture for Busemann non-positively curved spaces
• The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces
• Roe C*-algebra for groupoids and generalized Lichnerowicz Vanishing theorem for foliated manifolds

Cited by in corpus (3)

• Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
• The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions
• Crossed product approach to equivariant localization algebras