Injective maps between flip graphs (Q5962663)

In this paper, the authors study simplicial maps between \textit{flip graphs} \(\mathcal{F}(S)\) associated to surfaces \(S\). Flip graphs are also called \textit{triangulation graphs}, or \textit{ideal triangulation graphs}. The authors prove that if \(S\) is a surface with punctures (or marked points) of finite topological type which is complicated enough (the exceptions being the essential subsurfaces of the torus) and if \(S'\) is any other surface of finite topological type, then, every injective simplicial map \(\mathcal{F}(S)\to \mathcal{F}(S')\) is induced by an embedding \(S\to S'\). The theorem generalizes a theorem obtained in [\textit{M. Korkmaz} and \textit{A. Papadopoulos}, Ann. Inst. Fourier 62, No. 4, 1367--1382 (2012; Zbl 1256.32015)].NEWLINENEWLINENEWLINENEWLINEThe generalization is substantial for two reasons. First, the new result is a local (versus a global) theorem. Secondly, the proof of the local theorem is much more involved than the proof of the global theorem. It needs new subtle combinatorial arguments which the authors develop. Using a theorem in [\textit{V. Disarlo} and \textit{H. Parlier}, ``The geometry of flip graphs and mapping class groups'', preprint, \url{arXiv:1411.4285}], the authors deduce the following result: Let \(S\) be a surface satisfying the same hypotheses as above and \(\phi:\mathcal{F}(S)\to \mathcal{F}(S')\) a simplicial map. Then, \(\phi(\mathcal{F}(S))\) is totally geodesic in \(\mathcal{F}(S')\).NEWLINENEWLINENEWLINENEWLINEIn their proofs, the authors use techniques they introduced in the paper [\textit{J. Aramayona}, Geom. Dedicata 144, 115--128 (2010; Zbl 1194.57020)].NEWLINENEWLINENEWLINENEWLINEThe reviewer would like to add that although there are several rigidity results on mapping class groups of surfaces in which the statements have the same flavor, many of these results are individually important, because the settings are often completely different. The methods used and the techniques developed for the proofs are different, and the applications are different.NEWLINENEWLINENEWLINENEWLINEThe study of flip graphs is now increasingly important. These graphs appear for instance in the joint work of W. Thurston with computer scientists, and in more recent works of D. Thurston, Costantino, Martelli, Penner, Parlier, and in several works of applied mathematicians.