# Number of fixed order elements in the mapping class group of a closed surface

Let $S_g$ be a closed surface of genus $g\geq 2$. Given $r \in \mathbb{N}$, what is the number of elements of order $r$ in the mapping class group? Is it finite or infinite? If it is infinite is there any way to generate such a class? If it is finite is there an upper bound on the number? Any reference of link will be extremely helpful. Thanks in advance.

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## migrated from mathoverflow.netFeb 6 '14 at 22:16

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The number should always be $\infty$ or $0$. Think about a 180-degree rotation: this provides one order-2 element. But all of its conjugates (i.e. mapping classes which look in some coordinates like a 180-degree rotation) will also have order 2. Except for the hyperelliptic involution in genus 2, I believe every finite-order element of the mapping class group should have infinitely many conjugates (hence the claim, since there are other involutions in genus 2). Depending on your application, a better question might be "How many conjugacy classes are there of elements of order $r$?" – Tom Church Feb 6 '14 at 15:10
Centralizers in these groups are almost always (infinite) cyclic, no? That should immediately give infinitely many conjugates. – Steve D Feb 6 '14 at 17:27
I think you should revise your question to ask for conjugacy classes of finite-order elements, of which there are finitely many, and they are enumerable. Otherwise, I would vote to close the question or shift it over to math.stackexchange – Agol Feb 6 '14 at 17:28
Please see, for example, "A primer on mapping class groups", by Farb and Margalit. I've voted to close, by moving the question. – Sam Nead Feb 6 '14 at 17:49