Essential circles and Gromov-Hausdorff convergence of covers (Q2797815)

Following Spanier's classical construction, Sormani and Wei constructed\(,\) for \(\delta>0,\) a covering map \(\pi^{\delta}:\widetilde{X}^{\delta}\rightarrow X\) over a geodesic space \(X\) using an open cover of \(X\) consisting of \(\delta \)-balls. Later on Berestovskii and Plaut constructed covering maps over uniform spaces \(X\) using discrete chains and homotopies. Particularly\(,\) if \(X\) is a connected metric space and \(\varepsilon>0\), the Berestovskii and Plaut construction yields an \(\varepsilon\)-covering map \(\varphi_{\varepsilon}:X_{\varepsilon }\rightarrow X\) with deck group \(\pi_{\varepsilon}(X).\) The group \(\pi_{\varepsilon}(X)\) can be considered as a kind of fundamental group of \(X\) at a given scale. The authors show that\(,\) for a geodesic space \(X\) and \(\delta=\frac{3\varepsilon}{2},\) the covering maps \(\pi^{\delta}\) and \(\varphi_{\varepsilon}\) over \(X\) are naturally isometrically equivalent which implies that the covering spaces \(X_{\varepsilon}\) and \(\widetilde{X}^{\delta }\) share the same metric and topological properties. \textit{C. Sormani} and \textit{G. Wei} [Trans. Am. Math. Soc. 353, No.9, 3585--3602 (2001; Zbl 1005.53035)] noticed that \(\delta\)-covers over a geodesic space are not ``closed'' with respect to Gromov-Hausdorff convergence. That means the following: if a sequence \((X_{n})\) of compact geodesic spaces \(X_{n}\) converges to a compact space \(X\) in the Gromov-Hausdorff sense\(,\) the sequence \(((X_{n})_{\varepsilon})\) of related \((\)possibly non-compact\()\) covering spaces \((X_{n})_{\varepsilon}\) may not converge to \(X_{\varepsilon}\) in the pointed Gromov-Hausdorff sense. Even if the sequence \(((X_{n})_{\varepsilon})\) converges\(,\) its limit may not be an \(\varepsilon\)-covering space over \(X.\) To overcome this fault the authors extend the notion of \(\varepsilon\)-covering map over a compact geodesic space \(X\) to the notion of circle \((\mathcal{T},\varepsilon)\)-covering map \(\varphi_{\varepsilon}^{\mathcal{T}}:X_{\varepsilon}^{\mathcal{T}}\rightarrow X.\) Here \(\mathcal{T}\) denotes a finite \((\)possibly empty\()\) collection of essential circles in \(X\) having lengths at least \(3\delta,\) where \(\delta \geq\varepsilon.\) The essential circles\(,\) introduced by the authors in a previous paper [Adv. Math. 232, No. 1, 271--294 (2013; Zbl 1260.57039)], are a special kind of closed geodesic characterized as being as short as possible in a discrete version of its free homotopy class. A circle covering map \(\varphi_{\varepsilon}^{\mathcal{T}}:X_{\varepsilon }^{\mathcal{T}}\rightarrow X\) is obtained in the following way. The collection \(\mathcal{T}\) naturally determines a normal subgroup \(K_{\varepsilon }(\mathcal{T})\) of the deck group \(\pi_{\varepsilon}(X)\) of \(\varphi _{\varepsilon}:X_{\varepsilon}\rightarrow X,\) where \(K_{\varepsilon} (\emptyset)\) is the trivial group by definition. The subgroup \(K_{\varepsilon }(\mathcal{T})\) acts freely and discontinuously on \(X_{\varepsilon}\) with quotient \(X_{\varepsilon}^{\mathcal{T}}.\) A natural induced mapping \(\varphi_{\varepsilon}^{\mathcal{T}}:X_{\varepsilon}^{\mathcal{T}}\rightarrow X\) is a covering map with the deck group naturally isomorphic to \(\pi_{\varepsilon}^{\mathcal{T}}(X)=\pi_{\varepsilon}(X)\left/ K_{\varepsilon }(\mathcal{T})\right. .\) Particularly\(,\) for \(\mathcal{T}=\emptyset,\) \(\varphi_{\varepsilon}^{\mathcal{\emptyset}}=\varphi_{\varepsilon}\) and \(\pi_{\varepsilon}^{\emptyset}(X)=\pi_{\varepsilon}(X).\) Then the authors prove the following main result about circle covering maps. Let \((X_{n})\) be a sequence of compact geodesic spaces \(X_{n}\) which converges to a compact geodesic space \(X\) in the Gromov-Hausdorff sense and let \(((\varphi_{n})_{\varepsilon_{n}} ^{\mathcal{T}_{n}})\) be a sequence of circle covering maps \((\varphi_{n})_{\varepsilon_{n}}^{\mathcal{T}_{n}}:(X_{n})_{\varepsilon_{n} }^{\mathcal{T}_{n}}\rightarrow X_{n},\) where, for each \(n\in\mathbb{N},\) \(\mathcal{T}_{n}\) is a finite collection of essential circles in \(X_{n}\) of lengths \(\delta\geq \varepsilon_{n}\) and the sequence \((\varepsilon_{n})\) has a positive lower bound. Then, for each \(\varepsilon,0<\varepsilon \leq\lim\inf(\varepsilon_{n}),\) there exists a subsequence \((X_{n_{k}})\) of \((X_{n})\) and a finite collection \(\mathcal{T}\) of essential circles in \(X\) of lengths \(\delta\geq \varepsilon\) such that a sequence \(((X_{n_{k} })_{\varepsilon_{n_{k}}}^{\mathcal{T}_{n_{k}}})_{k}\) of circle covering spaces \((X_{n_{k}})_{\varepsilon_{n_{k}}}^{\mathcal{T}_{n_{k}}}\) converges to a circle covering space \(X_{\varepsilon}^{\mathcal{T}}\) in the pointed Gromov-Hausdorff sense and the group \(\pi_{\varepsilon_{n_k}}^{\mathcal T_{n_{k}}}(X_{n_{k}})\) is isomorphic to \(\pi_{\varepsilon}^{\mathcal{T}}(X),\) for all large \(k.\)