Generating the surface mapping class group by two elements

Abstract

Abstract. Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.

Promoting circular-orderability to left-orderability

“…3 surface with one puncture (see [18] for an introduction to mapping class groups), which is circularly-orderable but not left-orderable since it contains torsion. Furthermore, since G is generated by torsion elements [28], it has no non-trivial left-orderable quotients. By a result of Harer [20], G/G = {id}.…”

A note on the generating sets for the mapping class groups

“…Then Korkmaz improved this result, by the following theorem: Theorem 1. [4] Suppose that g 2 and g is a closed oriented surface of genus g. The mapping class group MCG ( g ) of g is generated by S and B.…”

Generating the Mapping Class Group of a Punctured Surface by Involutions

“…Brendle and Farb [4] answered Luo's question by constructing a generating set of M g,0 consisting of three elements of order 2g + 2, 4g + 2 and 2 (or g) for g ≥ 1. Korkmaz [14] showed that M g,0 is generated by two torsion elements, each of order 4g + 2 for g ≥ 3 and p = 0, 1, and therefore his torsion generating set is minimal. The author [24] proved that M g,0 is generated by three elements of order three and by four elements of order four for g ≥ 3.…”

“…Yoshihara [31] showed that M g,0 is generated by three elements of order 6 if g ≥ 10 and by four elements of order 6 if g ≥ 5. Yildiz [30] improved the result of [14] by showing that M g,0 is generated by two elements of order g for g ≥ 6. From the above results, it is natural to ask the following: Question 5.2.…”