0 Torsion Elements in the Mapping Class Group of a Surface

Given a finite set of r points in a closed surface of genus g, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if (g, r) = (2, 5k + 4) for some integer k. §1. Introduction 1.1. The purpose of this paper is to investigate when the mapping class group of a compact surface is generated by torsion elements. Our result gives a complete answer to this question. Theorem. Suppose Σ g,r is a compact orientable surface of genus g with r many boundary components where g, r ≥ 0. Then the mapping class group of the path components of the orientation preserving homeomorphisms of the surface is generated by torsion elements if and only if (g, r) = (2, 5k + 4) for some integer k ∈ Z. The torsion elements in the mapping class group of the surface Σ 2,5k+4 generate an index 5 subgroup. Furthermore, in the case of (g, r) = (2, 5k + 4), the order n of the torsion elements generating the group can be chosen as follows: (a) if g ≥ 3, then n = 2; Note that by identifying each boundary component to a point, one sees that the mapping class group in the theorem is the same as the mapping class group of a closed surface leaving a set of r points invariant. The existence of the exceptional cases (2, 5k + 4) is caused by the fact that there is only one non-trivial Z 5-action on the closed surface of genus-2. And this Z 5-action has too few fixed points (only 3 points). The result is motivated by the example of the torus whose mapping class group is SL(2, Z) = Z 4 * Z 2 Z 6. In the case of surfaces of genus at least 3, the theorem can be derived easily from the work of Harer [Ha]. A simple derivation of it will be given in §1.3. The main body of the paper is to prove the theorem for surfaces of genus at most 2. On the related question of Torelli groups, the following can be derived easily from the work of Johnson [Jo] and Powell [Po]. Namely the Torelli group of a closed surface of genus at least 3 is generated by …