The equivariant coarse Novikov conjecture and coarse embedding

TL;DR

This paper proves the equivariant coarse Novikov conjecture for spaces with group actions that admit coarse embeddings into Hilbert space, under certain bounded distortion and geometric conditions.

Contribution

It establishes the conjecture's validity for spaces with group actions satisfying coarse embeddability and bounded distortion conditions.

Findings

Proves the equivariant coarse Novikov conjecture under specified conditions.

Shows that coarse embeddability into Hilbert space implies the conjecture holds.

Provides conditions for nonvanishing of equivariant higher indices.

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 $X$.