Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: General primitivity in the mapping class group   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 $\mathrm{Mod}(S_g)$.

31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex