Stable specific torsion length and periodic mapping classes (Q2140637)

Let \(S_g\) be a closed orientable surface of genus \(g \geq 3\) and let \(\mathrm{Mod}(S_g)\) denote its mapping class group. This article concerns periodic mapping classes in \(\mathrm{Mod}(S_g)\). Let \(\phi \in \mathrm{Mod}(S_g)\) be a periodic mapping class of order at least \(2\). Theorem 1.1 in this article shows that a suitable power of \(\phi\) maps a non-separating simple closed curve \(c\) on \(S_g\) to a distinct disjoint curve \(d\). The authors prove Theorem 1.1 considering the following cases. When \(\phi\) acts freely on \(S_g\), it acts as a covering transformation and the assumption \(g \geq 3\) produces such a \(c\). When \(\phi\) is an order two element and has a fixed point, they use \textit{F. Klein}'s [Math. Ann. 42, 1--29 (1893; JFM 25.0689.03)] classification of order \(2\) homeomorphisms of \(S_g\) to produce such a \(c\). When \(\phi\) has order at least three and has a fixed point, they use \textit{R. S. Kulkarni}'s [Contemp. Math. 201, 63--79 (1997; Zbl 0863.30050)] description of \(S_g\) as a quotient space of a regular \(n\)-gon acted upon by \(\phi\) as a rotation of this \(n\)-gon to produce such a curve \(c\). It is in this latter case that a certain power of \(\phi\), if not \(\phi\) itself, can achieve the disjointness of the curves \(c\) and \(d\). The authors further show that except when \(\phi\) is the hyperelliptic involution, the curve \(c\) can be chosen so that \(c\) and \(d\) do not form a bounding pair. Let \(T_{\alpha}\) denote the Dehn twist about a non-separating simple closed curve \(\alpha\) on \(S_g\). The authors use the lantern relation in \(\mathrm{Mod}(S_g)\) and Theorem 1.1 to show that \(T_{\alpha}\) can be expressed as a product of six conjugates of \(\phi\). Using this they conclude that if \(t\) is any torsion element in \(\mathrm{Mod}(S_g)\), then for a suitable \(n\), the specific stable torsion length with respect to \(t^n\) of \(T_{\alpha}\) is bounded above by \(6\). Throughout the article, the authors pose interesting questions related to mapping classes in \(Mod(S_g)\) and to specific stable torsion lengths.