On translation lengths of pseudo-Anosov maps on the curve graph (Q6589623)

Let \(S:=S_{g,n}\) denote a connected, oriented surface of genus \(g\) and \(n\) punctures. The mapping class group, denoted by \(\text{Mod}(S)\), is defined as a group of orientation-preserving self homeomorphisms up to isotopy. Elements of \(\text{Mod}(S)\) are known as mapping classes. Let \(\mathcal{C}(S)\) denote the curve graph, whose vertices are equivalence classes of isotopic essential simple closed curves in \(S\) and two vertices have an edge if their classes have disjoint representatives. Closed curves \(a_1, a_2,\ldots, a_k\) are called filling if \(S\setminus \{a_1, a_2,\ldots, a_k\}\) is a disjoint union of disks or punctured disks. It is a well-known fact that \(\text{Mod}(S)\) has a natural action on \(\mathcal{C}(S)\). A mapping class \(f\) is called pseudo-Anosov if no power of \(f\) fixes any essential simple closed curve. The asymptotic translation length of the mapping class \(f\) is given by, \N\[\Nl_{\mathcal{C}}(f):=\text{lim}_{n\to \infty} \frac{d_{\mathcal{C}}(a,f^n(a))}{n}.\N\]\NIn the paper under review, using two filling curves \(a,b\) such that \(d_{\mathcal{C}}(a,b)\geq 3\), the authors construct a family of pseudo-Anosov maps that preserve geodesic axes and they explicitly calculate their asymptotic translation lengths (Theorem 3.1). They also give bounds on asymptotic translation lengths for those pseudo-Anosov maps, which are constructed using more than two filling curves. As applications of the main result the authors prove the following results:\N\begin{itemize}\N\item[1.] Among the specific words of pseudo-Anosov maps, they determine which words have the smallest translation length on the curve graph.\N\item[2.] They construct a new class of pseudo-Anosov maps that optimizes the ratio of stable translation lengths on the curve graph to that on Teichmüller space.\N\item[3.] They give conditions on \(n\) and a collection of curves \(\{a_1, a_2,\ldots, a_t\}\) under which \(\{T_{a_1}^n, T_{a_2}^n,\ldots, T_{a_t}^n\}\) forms a right-angled Artin group.\N\end{itemize}