WoG 2024 Talk 3.3: Pankaj Kapari - Primitivity of pseudo-periodic mapping classes

Speaker: Pankaj Kapari
Institution: IISER Bhopal
Website: https://sites.google.com/view/pankajk...

Title: Primitivity of pseudo-periodic mapping classes
Abstract: For $g \ge 2$, let Mod$(S_g)$ be the mapping class group of the closed
oriented surface $S_g$ of genus $g$. A nontrivial $G \in$ Mod$(S_g)$ is said to
be a root of $F \in$ Mod$(S_g)$ of degree $n$ if there exists an integer $n \gt 1$
such that $G^n = F$ and $|G| = n|F|$. If $F$ does not have any roots,
then it is said to be primitive. A natural question is whether one can
determine if an arbitrary $F \in$ Mod$(S_g)$ is primitive and compute the
roots of $F$ (up to conjugacy) when it is not primitive. We call this the
general primitivity problem in Mod$(S_g)$. To begin with, we provide a solution to this problem for reducible mapping classes of infinite order. Using this solution, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we formulate a theoretical algorithm for solving the general primitivity problem in Mod$(S_g)$.Then we discuss realizable bounds on the degree of roots of reducible mapping classes in Mod$(S_g)$, the Torelli group $I(S_g)$, and the level $m$ congruence subgroup Mod$(S_g)[m]$ of Mod$(S_g)$. We conclude the talk with a result on normal closure of pseudo-periodic mapping classes in Mod$(S_g)$.