Products of Farey graphs are totally geodesic in the pants graph (Q2799113)

Let \(\Sigma=\Sigma_{g,b}\) be a compact connected orientable surface of genus \(g\) with \(b\) boundary components and \(\xi(\Sigma):=3g+b-3>0\); we call \(\xi(\Sigma)\) the complexity of \(\Sigma\).NEWLINENEWLINEConsider a multicurve \(Q\) on \(\Sigma\). Let \(\Sigma_Q\) be the complementary subsurface of \(Q\) in \(\Sigma\), consisting of the components of \(\Sigma\setminus \eta(Q)\) which are not pairs of pants; here \(\eta(Q)\) is an open regular neighborhood of \(Q\). The multicurve \(Q\) is called an \((n\times 1)\)-multicurve if \(\Sigma_Q\) consists of \(n\) components and every component has complexity \(1\).NEWLINENEWLINEDenote by \(\mathcal{P}(\Sigma)\) the pants graph of \(\Sigma\) and by \(\mathcal{P}_Q(\Sigma)\) the full subgraph of \(\mathcal{P}(\Sigma)\) consisting of all pants decompositions containing \(Q\). The Main Theorem of the paper states that if \(Q\) is an \((n\times 1)\)-multicurve then \(\mathcal{P}_Q(\Sigma)\) is totally geodesic in \(\mathcal{P}(\Sigma)\). This allows to give, in a partial case, a positive answer to a question by \textit{J. Aramayona} et al. [Math. Res. Lett. 15, No. 2--3, 309--320 (2008; Zbl 1148.57004)] who conjectured that for an injective simplicial map \(i:\mathcal{P}(\Sigma')\to \mathcal{P}(\Sigma)\) the image \(i(\mathcal{P}(\Sigma'))\) is totally geodesic in \(\mathcal{P}(\Sigma)\).NEWLINENEWLINEThe Main Theorem has some interesting implications. Thus, Corollary 1.1 claims that if \(G\) is a graph product of Farey graphs and \(\varphi:G\to \mathcal{P}(\Sigma)\) is an injective simplicial map, then \(\varphi(G)\) is totally geodesic in \(\mathcal{P}(\Sigma)\). Corollary 1.2 states that an isometric embedding \(i:\mathbb{Z}^r\to \mathcal{P}(\Sigma)\) exists if and only if \(r\leq [(3g+b-2)/2]\). Corollary 1.3 describes all such embeddings for the case \(r=(3g+b-2)/2\in \mathbb{N}\).NEWLINENEWLINEAs the authors check in the introduction of the paper, \textit{J. L. Estévez} [``Large flats in the pants graph'', Preprint, \url{arXiv:1306.3170}] announced that the Main Theorem holds if, in addition, one assumes that each boundary component of \(\Sigma_Q\) is separating, and as a consequence, he gives an alternative proof of Corollary 1.2.