The curve complex has dead ends (Q2516419)

Let \(\Sigma\) be an orientable surface of genus \(g\) with \(n\) boundary components and \(3g-3+n>0\). The curve graph \(\mathcal{C}^1(\Sigma)\) of \(\Sigma\) is a graph whose vertices are represented by homotopy classes of simple closed curves in \(\Sigma\). The graph \(\mathcal{C}^1(\Sigma)\) can be given the edge-length metric and studied as a metric space. The study of the large-scale geometry of the curve graph is complicated by two local pathologies: each vertex shares an edge with infinitely many other vertices and (typically) there are infinitely many geodesics joining two vertices. The paper under review points out a further yet-unnoticed pathology: dead ends exist in \(\mathcal{C}^1(\Sigma)\) and yet all have depth \(1\). Thus, there are vertices \(\alpha\) and \(\beta\) so that no geodesic joining \(\alpha\) to \(\beta\) can be extended to a geodesic beyond \(\beta\) (\(\beta\) is a dead end), yet if the geodesic is shortened by length \(1\) it can be extended to a geodesic of length at least \(2\) (it has depth \(1\)). Double dead ends also exist.