Roots of Dehn twists about separating curves (Q2873595)

Let \(C\) be a separating simple closed curve in a closed oriented surface \(F\) of genus \(g\geq 2\), and let \(n\geq 2\) be an integer. An \textit{\(n\)-th root} of the Dehn twist \(t_C\) about \(C\) is an element \(f\) of the mapping class group of \(F\) such that \(f^n=t_C\). The paper under review gives a combinatorial description of the set of (conjugacy classes of) \(n\)-th roots of \(t_C\).NEWLINENEWLINEFor this, the author considers the closed oriented surfaces \(F_1\) and \(F_2\) of genus \(g_1\) and \(g_2\) respectively (with \(g=g_1+g_2\) and \(g_1 \geq g_2\)) which result from cutting \(F\) along \(C\) and gluing a disk to each piece. A \textit{nestled \((n_i,\ell_i)\)-action} on the surface \(F_i\) is an action of the cyclic group \(\mathbb{Z}/n_i \mathbb{Z}\) on \(F_i\) such that the fixed points of some non-trivial element of \(\mathbb{Z}/n_i \mathbb{Z}\) generate \(\ell_i+1\) orbits, one of which reduces to a point \(P_i\) fixed by all elements. The author shows that (up to equivalences) an \(n\)-th root of \(t_C\) corresponds to a nestled \((n_1,\ell_1)\)-action on \(F_1\) and a nestled \((n_2,\ell_2)\)-action on \(F_2\) such that \(n=\text{lcm}(n_1,n_2)\) and the ``rotation angles'' about the distinguished fixed points \(P_1,P_2\) add up to \(2\pi/n\) modulo \(2\pi\). Next, using orbifold theory, nestled \((n_i,\ell_i)\)-actions on \(F_i\) are shown to be combinatorially encoded by \textit{data sets}, which consist of finitely many integers and congruence classes of integers satisfying certain elementary arithmetic identities. The above condition on rotation angles of nestled actions easily translates into an arithmetic condition on the corresponding data sets, which provides a neat description of all the (conjugacy classes of) \(n\)-th roots of \(t_C\).NEWLINENEWLINEThe paper gives several applications of this combinatorial description of roots of Dehn twists. For instance, there always exists an \(n\)-th root of \(t_C\) if \(n=\text{lcm}(4g_1,4g_2+2)\) and an \(n\)-th root of \(t_C\) can exist only for \(n\leq 4g^2+2g\). A complete list of all conjugacy classes of \(n\)-th roots of \(t_C\) is also provided in the case \(g=2\) (where \(n\) can be \(2,3,4,6\) or \(12\)) and in the case \(g=3\) (where \(n\) can be \(2,3,4,5,6,8,10,12,15,20,24\) or \(30\)).