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 $X$.