### References & Citations

# Mathematics > Geometric Topology

# Title: General primitivity in the mapping class group

(Submitted on 16 Sep 2021 (v1), last revised 30 Sep 2021 (this version, v2))

Abstract: For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we obtain necessary and sufficient conditions under which a given pseudo-periodic mapping class can be a root of another up to conjugacy. Using this characterization, the canonical decomposition of (non-periodic) mapping classes, and some known algorithms, we give an efficient algorithm for computing roots of arbitrary mapping classes up to conjugacy. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in $\mathrm{Mod}(S_g)$, the Torelli group, the level-$m$ subgroup of $\mathrm{Mod}(S_g)$, and the commutator subgroup of $\mathrm{Mod}(S_2)$. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class $F$ is $3q(F)(g+1)(g+2)$, where $q(F)$ is a unique positive integer associated with the conjugacy class of $F$. Moreover, this bound is realized by the roots of the powers of Dehn twist about a separating curve of genus $[g/2]$ in $S_g$. Finally, for $g\geq 3$, we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates $\mathrm{Mod}(S_g)$. Consequently, we establish that there exist roots of bounding pair maps and powers of Dehn twists that normally generate $\mathrm{Mod}(S_g)$.

## Submission history

From: Kashyap Rajeevsarathy [view email]**[v1]**Thu, 16 Sep 2021 12:43:44 GMT (47kb,D)

**[v2]**Thu, 30 Sep 2021 10:26:11 GMT (91kb,D)

Link back to: arXiv, form interface, contact.