2021/09/16 by Pankaj Kapari, Kashyap Rajeevsarathy, Kapari, Pankaj +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary 57K20 #Secondary 57M60
paper · pdf · doi:10.48550/arxiv.2109.07937
openalex publication_date 2021/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 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 a root of a power of a Dehn twist about a separating curve of genus [g/2] in Sg, where g≡ 0,9 \pmod12. 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 Mod(Sg) is normally generated by a root of bounding pair map or a root of a nontrivial power of a Dehn twist.