Alladi Ramakrishnan Hall

On the fractional powers of Dehn twists

Kashyap Rajeevsarathy

IISER Bhopal

Let $S_g$ be a closed orientable surface of genus $g > 1$ and
let $t_C$ denote a left handed Dehn twist about a simple closed curve $C$ in
$S_g$. A \textit{fractional power} of $t_C$ of exponent $\ell/n$ is a $h$
in the mapping class group Mod$(S_g)$ such that $h^n = t_C^\ell$. When $\ell
=1$, $h$ is a simply a root of $t_C$. Since Mod$(S)$ is generated by
finitely many Dehn twists, it is an interesting endeavor to understand the
existence of such fractional powers. We will describe the general
geometric construction of fractional powers and also derive some
(elementary number-theoretic) conditions for their existence (in some
cases). We will also give an algebraic perspective to this problem and
indicate how it generalizes to the case of multicurves.