Separating pants decompositions in the pants complex (Q416781)

In the paper under review the author studies the net of pants decompositions of a surface \(S\) that contain a nontrivially separating curve in the entire pants complex \({\mathcal P}(S)\) and gives an approximation of the maximum distance in \({\mathcal P}(S)\) of any pants decomposition to a pants decomposition containing a nontrivially separating curve. Let \({\mathcal P}_{sep}(S)\) be the pants complex of separating curves. The author obtains the following theorem: Let \(S\) be a surface of genus \(g\) with \(n\) boundary components (or punctures) and set \(D_{g,n}=\max _{P\in {\mathcal P}(S)}(d_{{\mathcal P}(S)}(P, {\mathcal P}_{sep}(S))).\) Then, for any fixed number of boundary components (or punctures) \(n\), \(D_{g,n}\) grows asymptotically like the function \(\log (g)\), that is, \(D_{g,n}=\Theta(\log (g)).\) In particular, for closed surfaces of sufficiently large genus, \({{\log_{2} (2g+2)}\over 2} -2 \leq D_{g,0} \leq \{2 \log_{2} (g-1) +1\}.\) On the other hand, for any fixed genus \(g \geq 2\), and any \(n \geq 6g-5, D_{g,n}=2\). The proof uses the relation between the pants decomposition \(P\) in \({\mathcal P}(S)\) and its pants decomposition graph \(\Gamma (P)\). The lower bounds follows from an explicit constructive algorithm for an infinite family of high girth log-length connected graphs.