General primitivity in the mapping class group

Abstract

For g ≥ 2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg 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 Mod(Sg), the Torelli group, the level-m subgroup of Mod(Sg), and the commutator subgroup of Mod(S2). 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 Sg. Finally, for g ≥ 3, we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates Mod(Sg). Consequently, we establish that there exist roots of bounding pair maps and powers of Dehn twists that normally generate Mod(Sg).

Infinite metacyclic subgroups of the mapping class group

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