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 $X$.

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等变粗诺维科夫猜想和粗嵌入

等变粗诺维科夫猜想提供了一种算法，用于确定非紧流形上椭圆微分算子的等变高指数不为零。在本文中，我们证明了一定粗可嵌入性条件下的等变粗诺维科夫猜想。更准确地说，如果离散群 $\Gamma$ 正确地、等距地和有界失真地作用于有界几何空间 $X$，$X/\Gamma$ 和 $\Gamma$ 允许粗嵌入到希尔伯特空间中，那么 $ \Gamma$-equivariant 粗糙 Novikov 猜想对 $X$ 成立。这里有界失真意味着对于任何 $\gamma\in\Gamma$，$\sup_{x\in Y} d(\gamma x,x)<\infty$，其中 $Y$ 是 $\对 $X$ 的 Gamma$-action。