Dehn twists and products of mapping classes of Riemann surfaces with one puncture (Q1938741)

Let \(\Sigma\) be a Riemann surface of genus at least \(2\) with one puncture, say \(x\). Let \(\mathcal{S}\) be the collection of simple closed geodesics on \(\Sigma\), and let \(\mathcal{F}\) be the set of mapping classes on \(\Sigma \) isotopic to the identity on \(\Sigma\cup \{ x\}\). Let \(t_c\) be the positive Dehn twist about a curve \(c\in \mathcal{S}\). The author investigates the products of the form \(\left( t_b^{-m}\circ t_a^n\right)\circ f^k\), where \(a,b\in \mathcal{S}\) and \(f\in \mathcal{F}\). Note that if \(a=b\) or \(a,b\) are boundary components of an \(x\)-punctured cylinder on \(S\), then one can find an element \(f\in \mathcal{F}\) such that the sequence \(\left( t_b^{-m}\circ t_a^n\right)\circ f^k\) contains infinitely many powers of Dehn twists. In the paper, the author proves that the converse statement is also true. More precisely, if the sequence \(\left( t_b^{-m}\circ t_a^n\right)\circ f^k\) contains infinitely many powers of Dehn twists, then \(a,b\) must be the boundary components of an \(x\)-punctured cylinder on \(S\) and \(f\) is a power of the spin map \( t_b^{-1}\circ t_a^1\).