Small torsion generating sets for mapping class groups

Abstract

A surface of genus g has many symmetries. These form the surface’s mapping class group Mod(S_g), which is finitely generated. The most commonly used generating sets for Mod(S_g) are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that Mod(S_g) is generated by six involutions for g ≥ 3. We will discuss our extension of these results to elements of arbitrary order: for k > 5 and g sufficiently large, Mod(S_g) is generated by three elements of order k. Generalizing this idea, in joint work with Margalit we showed that for g ≥ 3 every non-trivial periodic element that is not a hyperelliptic involution normally generates Mod(S_g). This result raises a question: does there exist an N, independent of g, so that if f is a periodic normal generator of Mod(S_g), then Mod(S_g) is generated by N conjugates of f? We show that in general there does not exist such an N, but that there do exist universal bounds when additional conditions are placed on f.