We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that, for a fixed genus g, there are at most five such classes. | Turk J Math 28 (2004) , 101 – 110. ¨ ITAK ˙ c TUB Conjugacy Classes of Finite Subgroups of Certain Mapping Class Groups Michal Stukow Abstract We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that, for a fixed genus g, there are at most five such classes. Key words and phrases: Hyperelliptic mapping class group, finite subgroups of mapping class group. 1. Introduction Let M0,r be the mapping class group of the sphere with r ≥ 3 marked points, where we allow maps to permute the set of marked points. In [3] Gillette and Van Buskirk proved, using purely algebraic methods, the following theorem. Theorem 1 M0,r contains an element of finite order n if and only if n divides one of r, r − 1, r − 2. Later on the stronger version of this theorem was obtained as a by-product of certain considerations of Harvey and Maclachlan [5]. 1991 Mathematics Subject Classification: Primary 57M60; Secondary 57N05 Supported by BW5100-5-0227-2 101 STUKOW Theorem 2 Every element of finite order in M0,r is contained in a maximal cyclic subgroup of order r, r −1 or r −2, and all such subgroups of the same order are conjugate. In this paper we extend the above results. For every finite subgroup N of M0,r we find a maximal finite subgroup M of M0,r containing N . Furthermore, we give a complete description of conjugacy classes of finite subgroups of M0,r . As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. Maximal finite subgroups of M0,r 2. This section is devoted to the proof of the following theorem. Theorem 3 Finite subgroup N of M0,r is a maximal finite subgroup of M0,r if and only if N is isomorphic to one of the following: 1. the cyclic group Zr−1 if r 6= 4, 2. .
