Hausdorff convergence and universal covers (Q2716151)

Given two metric spaces \(X\) and \(Y\), the Gromov-Hausdorff distance between them is denoted by \(d_{GH}(X,Y)\), and is defined by the infimum of \(d_{h}^{Z}(f(X),f(Y))\) for all metric spaces \(Z\) and isometric embeddings \(f:X\rightarrow Z\), \(g:Y\rightarrow Z\) where \(d_{h}^{Z}\) is the Hausdorff distance between subsets of \(Z\). If \(\;x\in X\) and \(y\in Y\), the pointed Gromov-Hausdorff distance is denoted by \(d_{GH}((X,x),(Y,y))\), and defined as the infimum of \( d_{h}^{Z}(f(X),f(Y))\) for all metric spaces \(Z\) and isometric embeddings \(f:X\rightarrow Z\), \(g:Y\rightarrow Z\), such that \(f(x)=g(y)\). Non-compact length spaces \((X_{n},x_{n})\) converge in the Gromov-Hausdorff sense to \((Y,y)\), if for any \(R>0\) there exists a sequence \(\varepsilon _{n}\rightarrow 0\) such that \(B_{x_{n}}(R+\varepsilon _{n})\) converges to \(B_{x_{n}}(R+\varepsilon _{n})\) in the Gromov-Hausdorff sense. Such limit spaces of manifolds with lower bounds on Ricci curvature have been studied in recent years both from a geometric and a topological perspective. NEWLINENEWLINENEWLINEIn this paper the authors consider Riemannian manifolds without boundary and study Gromov-Hausdorff limits of sequences of them with a uniform upper bound on diameter and lower bound on Ricci curvature. \textit{J. Cheeger} and \textit{T. Colding} [J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)]\ proved that the limits of such sequences are locally connected at special points. It was also already shown that the limit space could locally have infinite topological type [see \textit{X. Menguy}, in ``Examples with bounded diameter growth and infinite topological type'', Duke Math. J. 102, No. 3, 403-412 (2000; Zbl 1163.53323)]. In this paper the authors prove that the universal cover of the Gromov-Hausdorff limit space exists.NEWLINENEWLINENEWLINEIt is also shown that if \(M_{i}\) is the sequence of manifolds under consideration, there exists for \(i\) large enough a surjective homeomorphism from the fundamental group of \(M_{i}\) onto the group of deck transforms of the Gromov-Hausdorff limit space. In the non-collapsed case, if the sequence of manifolds has uniform lower bound on the volume, it is shown that the kernels of these surjective homeomorphism are finite with a uniform bound on their cardinality. The article also includes some results about the limits of covering spaces and their deck transforms, when the sequence of manifolds is assumed to be of compact length spaces with a uniform upper bound on the diameter.NEWLINENEWLINENEWLINEThe authors begin studying limit spaces of compact length spaces with upper bounded limit that converge in the Gromov-Hausdorff sense to a limit space. In section 2, two examples of such sequences are given, however their fundamental group cannot be mapped surjectively onto the fundamental group or revised fundamental group of the limit space. In section three \(\delta\)-covering spaces are introduced. Given \(\delta >0\), the \(\delta\)-cover of a length space \(Y\), denoted by \(\widetilde{Y}^\delta\), is defined to be \(\widetilde{Y}_{{\mathcal{U}}_{\delta}}\) where \({\mathcal{U}}_{\delta}\) is the open covering of \(Y\) consisting of all balls of radius \(\delta\). It is shown that the limit of \(\delta\)-covers is a cover of the limit space. In the last section limit spaces of sequences of compact manifolds with a uniform upper bound on diameter and lower bound on Ricci curvature are studied and it is proved that the universal cover of the limit space exist.NEWLINENEWLINENEWLINEIn the end of the introduction, the authors give a list of references for background material in Gromov-Hausdorff limits and Ricci curvature, and for covering spaces and fundamental groups.