The boundary at infinity of the curve complex and the relative Teichmüller space (Q2102165)

Let \(S\) be a surface of a finite genus with finitely many punctures. Let \(\mathcal{T}(S)\) denote the Teichmüller space of \(S\), which is the space of all equivalence classes of conformal structures of finite type on the surface \(S\). A conformal structure is of finite type if each puncture has a neighborhood conformally equivalent to a punctured disk. Let \(\alpha\) be a homotopy class of simple closed curves on \(S\). Thin\(_{\alpha}\) denotes the region of \(\mathcal{T} (S)\); that is, Thin\(_{\alpha} = \{ \sigma \in \mathcal{T} (S) \mid \mathrm{ext}_{\sigma}(\alpha) \leq \epsilon \,\}\) for some fixed small \(\epsilon > 0\), where ext\(_{\sigma}(\alpha)\) is the extremal length of \(\alpha\). We note that the extremal length ext\(_{\sigma}(\alpha) = \sup_{\rho} \frac{(l_{\rho}(\alpha))^2}{A_{\rho}}\), where \(\sigma \in \mathcal{T}(S)\), \(\rho\) ranges over all metrics in the conformal class of \(\sigma\), \(l_{\rho}(\alpha)\) is the infimum of the length of \(\alpha\) with respect to \(\rho\) and \(A_{\rho}\) is the area of \(S\) with respect to \(\rho\). In this paper, the author aims to describe the boundary at infinity of the electric Teichmüller space \(\mathcal{T}_{el} (S)\), which is obtained from \(\mathcal{T} (S)\) by collapsing every region Thin\(_{\alpha}\) to diameter 1. To put it more explicitly, the author shows the following theorem: The boundary at infinity of \(\mathcal{T}_{el} (S)\) is homeomorphic to the space of minimal topological foliations on \(S\). Furthermore, the author shows that this result holds for the curve complex \(\mathcal{C} (S)\); that is, the boundary at infinity of \(\mathcal{C} (S)\) is the space of minimal topological foliations on \(S\). We point out that the curve complex \(\mathcal{C} (S)\) is a simplicial complex whose vertices are homotopy classes of non-peripheral simple closed curves on the surface \(S\). Here, two vertices are connected by an edge if curves corresponding to these vertices are realized disjointly on the surface \(S\).