Let $(\widetilde{M_{j}^{n}},\tilde{g}_{j},\Gamma_{j},\tilde{p}_{j})$ be the Riemannian universal covers of $(M_{j}^{n},g_{j})$ which converge to the limit product space $(\mathbb{R}^{n-1}\times Y,\tilde{d}_{\infty},\Gamma_{\infty},\tilde{p}_{\infty})$ in the equivariant Gromov-Hausdorff topology, where $\Gamma_{j}\equiv\pi_{1}(M_{j}^{n})$ and $\Gamma_{j}\to\Gamma_{\infty}\leq\Isom(\mathbb{R}^{n-1}\times Y)$. See Section 3 of [FY92] for the precise definition of the equivariant Gromov-Hausdorff convergence. In summary, we have the following diagram

where the covering maps $\pr_{j}:\widetilde{M_{j}^{n}}\to M_{j}^{n}$ converge to a natural projection map $\pr_{\infty}:\mathbb{R}^{n-1}\times Y\to\mathbb{R}^{n-1}$.

The main part is to prove the claim that $Y$ is isometric to $\mathbb{R}$.

Applying Cheeger-Colding’s quantitative splitting theorem (see [CC96]), the convergence assumption (7.41) implies that for any fixed $R>0$, there are harmonic splitting maps $\Phi_{j}\equiv(u_{j}^{(1)},\ldots,u_{j}^{(n-1)}):B_{10R}(p_{j})\to\mathbb{R}^{n-1}$ which realize the Gromov-Hausdorff maps such that

Let $\widetilde{\Phi}_{j}\equiv(\tilde{u}_{j}^{(1)},\ldots,\tilde{u}_{j}^{(n-1)})$ be the lifted harmonic functions on the universal covers, then the volume comparison theorem implies that

By the definition of the splitting maps, $Y$ is the Gromov-Hausdorff limit of the level sets of the lifted splitting maps $\widetilde{\Phi}_{j}^{-1}(0^{n-1})$. Since $\Gamma_{2}(p_{j})\leq\pi_{1}(M_{j}^{n})$ is of infinite order which acts on $\widetilde{M}_{j}^{n}$ isometrically and discretely, the limit space $Y$ must be non-compact.

On other hand hand, notice that $\widetilde{\Phi}_{j}^{-1}(0^{n-1})$ is invariant under the deck transformation group $\Gamma_{j}$ and $\Gamma_{j}$ converge to some limiting group $\Gamma_{\infty}\leq\Isom(\mathbb{R}^{n-1}\times Y)$ such that $\pr_{\infty}$ is given by the quotient $(\mathbb{R}^{n-1}\times Y)/\Gamma_{\infty}=\mathbb{R}^{n-1}$. Hence $\Gamma_{\infty}\leq\Isom(Y)$ and $\Gamma_{\infty}$ acts homogeneously on $Y$.

Therefore, by standard arguments, the noncompact homogeneous space $Y$ admits a line (see [CG72] or lemma 2.4 in [NZ16]). The Ricci curvature assumption implies that $Y$ is isometric to $\mathbb{R}$. This completes the proof of the claim.

The curvature estimate (7.42) immediately follows from the $\epsilon$-regularity theorem for noncollapsed Einstein manifolds (for example see Section 7 in [CC97]).

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