General primitivity in the mapping class group

Abstract

For [Formula: see text], let [Formula: see text] be the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text]. 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 algorithm for determining the conjugacy classes of roots of arbitrary mapping classes. Furthermore, we derive realizable bounds on the degrees of roots of pseudo-periodic mapping classes in [Formula: see text], the Torelli group, the level-[Formula: see text] subgroup of [Formula: see text], and the commutator subgroup of [Formula: see text]. In particular, we show that the highest possible (realizable) degree of a root of a pseudo-periodic mapping class [Formula: see text] is [Formula: see text], where [Formula: see text] is a unique positive integer associated with the conjugacy class of [Formula: see text]. Moreover, this bound is realized by a root of a power of a Dehn twist about a separating curve of genus [Formula: see text] in [Formula: see text], where [Formula: see text]. Finally, for [Formula: see text], we show that any pseudo-periodic mapping class having a nontrivial periodic component that is not the hyperelliptic involution, normally generates [Formula: see text]. Consequently, we establish that [Formula: see text] is normally generated by a root of bounding pair map or a root of a nontrivial power of a Dehn twist.