##### Venue: Department of Mathematics, Lecture Hall I

Let $F$ be a closed orientable surface of genus $g \\geq 2$ and $C$ be a simple closed curve in $F$. Let $t_C$ denote a left-handed Dehn twist about $C$. When $C$ is a nonseparating curve, D. Margalit and S. Schleimer showed the existence of such roots by finding elegant examples of roots of $t_C$ whose degree is $2g + 1$ on a surface of genus $g + 1$. This motivated an earlier collaborative work with D. McCullough in which we derived conditions for the existence of a root of degree $n$. We also showed that Margalit-Schleimer roots achieve the maximum value of $n$ among all the roots for a given genus. Suppose that $C$ is a separating curve in $F$. First, we derive algebraic conditions for the existence of roots in Mod$(F)$ of the Dehn twist $t_C$ about $C$. Finally, we show that if $n$ is the degree of a root, then $n \\leq 4g^2 + 2g$, and for $g \\geq 10$, $n \\leq \\frac{16}{5}g^2+ 12g + \\frac{45}{4}$.

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Last updated: 06 Mar 2020