The restriction of efficient geodesics to the nonseparating complex of curves (Q6607988)

The complex of curves \(C(S)\) for a compact surface \(S\) is a simplicial complex whose vertices correspond to isotopy classes of essential simple closed curves in \(S\) and whose edges connect vertices with disjoint representatives. If \(S\) has genus at least 2 then \(C(S)\) is connected. The vertices of \(C(S)\) may be given a metric \(d\) by defining the distance between two vertices \(u\) and \(v\) to be the minimal number of edges among paths between them.\N\NBirman, Margalit, and Menasco introduced a preferred finite set of geodesics, called \textit{efficient geodesics} in [\textit{J. Birman} et al., Math. Ann. 366, No. 3--4, 1253--1279 (2016; Zbl 1350.05022)]. They can be found by an algorithm. Let \(N(S)\) denote the subcomplex of \(C(S)\) spanned by vertices corresponding to nonseparating simple closed curves. Again, it is a standard result that if the genus of \(S\) is at least two, \(N(S)\) is connected.\N\N\textbf{Theorem 1.1.} Let \(S\) be a surface of genus at least two. If \(v\) and \(w\) are vertices of \(N(S)\) with \(d(v,w) \ge 3\), then there exists an efficient geodesic from \(v\) to \(w\) in \(N(S)\). Additionally, there are finitely many efficient geodesics from \(v\) to \(w\).