Strip maps of small surfaces are convex (Q2358262)

Let \(S\) be a compact surface with nonempty boundary and \(N = 6g - 6 + 3k\) where \(g\) is the genus of \(S\) and \(k\) the number of its boundary components; as it is often the case the paper under review considers only the case where \(N > 0\), or in other words \(\chi(S) < 0\). The arc complex \(\overline X\) of \(S\) is an \((N-1)\)-dimensional simplicial complex which encodes the combinatorics of arcs on \(S\). Its vertices are isotopy classes of arcs not contained in the boundary and simplices correspond to collections of arcs with pairwise disjoint representatives. Removing cells whose vertices do not decompose \(S\) into discs yields an open dense subset \(X \subset \overline X\). It is known that \(X\) is homeomorphic to a \((N-1)\)-ball [\textit{J. L. Harer}, Invent. Math. 84, 157--176 (1986; Zbl 0592.57009) and \textit{R. C. Penner}, Commun. Math. Phys. 113, 299--339 (1987; Zbl 0642.32012)]. One can then attempt to study the triangulation given by the combinatorial structure of \(X\). To do this some geometry is needed: using a hyperbolic structure on \(S\) it is possible to construct an embedding \(X \to \mathbb P(\mathbb R^N)\) whose image is a convex subset (this is the strip map from the title of the paper under review). The conjecture this paper addresses is whether the triangulation of this convex set can be straightened to a triangulation by convex subspaces. This is proven to be the case when \(S\) is ``smallest possible'', that is \(S\) is a sphere with three boundary components or a torus with one boundary component. In both cases the arc complex is fairly well-understood: in the former it is just the barycentric subdivision of a triangle. In the latter it is the Farey triangulation of the disc. Thus, as the author remarks, the interest of the paper resides more in the technical details of the proof, which might lead to a better understanding of the higher-complexity surfaces where there is no reasonably simple description of the arc complex.