# Conjugacy classes of the mapping class group

I am not sure if this is a well known problem, but I was not able to find anything online that answered my question.

Is it known how to tell whether two elements of the mapping class group of a surface are conjugate?

• I'd be curious whether considering conjugation in $\mathrm{MCG}^+$ (group of oriented self-diffeotopies) is equivalent to conjugation (of elements of $\mathrm{MCG}^+$) in $\mathrm{MCG}^\pm$ (the group of all self-diffeotopies, which is also $\mathrm{Out}(\Gamma_g)$ for a closed surface of genus $g$). For instance this fails in the $2$-torus, since a non-trivial unipotent matrix is conjugate to its inverse in $\mathrm{GL}_2(\mathbf{Z})$, but not in $\mathrm{SL}_2(\mathbf{Z})$.
– YCor
Apr 10, 2020 at 15:28

An exponential-time solution to the conjugacy problem in the mapping class group was given by Jing Tao, in:

Tao, Jing(1-OK) Linearly bounded conjugator property for mapping class groups. (English summary) Geom. Funct. Anal. 23 (2013), no. 1, 415–466.

Tao's main contribution is to prove that two conjugate periodic mapping classes have a conjugator of linear length; the pseudo-Anosov case had already been done by Mazur--Minsky.

To actually determine conjugacy in practice, I think the state of the art in this area is the recent papers of Bell--Webb, who show how to compute distance in the curve graph and Nielsen--Thurston type in practice. Bell has even implemented some of their algorithms (see here). See also the parallel results of Birman--Margalit--Menasco.

Added (4 March 2020): Mark Bell pointed out to me that his software can indeed effectively solve the conjugacy problem in specific examples. Flipper decides conjugacy of pseudo-Anosovs, while Curver decides conjugacy of periodic automorphisms.

Yes. The mapping class group is bi-automatic,so its conjugacy problem is decidable, https://arxiv.org/abs/0912.0137. For Ursula-skeptics, here is a much earlier paper solving the conjugacy problem G. Hemion, On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds, Acta Math. 142 (1979), no. 1-2, 123–155. If you are interested only in pA elements , then look at H. Masur and M. Minsky, Geometry of the complex of curves II: Hierarchical structure, GAFA, Geom. Funct. Anal., Vol. 10 (2000), 902–974.

• That paper has been on the arXiv for 10 years, and is still not published. This is a strong indication that, at the very least, it will be difficult to apply its results in practice. (The wording of the question strongly suggests that the OP wants to determine conjugacy of particular mapping classes, rather than just know abstractly that the conjugacy problem is solvable.)
– HJRW
Jan 1, 2020 at 12:00
• Without going into details, Mark, I can assure you that there are researchers in the field who regard the biautomaticity of mapping class groups as open. I'll also note that neither of the papers that you mention claims that mapping class groups are known to be biautomatic, and indeed, Farb implies on page 24 of his article that the question is open.
– HJRW
Jan 1, 2020 at 19:08
• It's an important result, so probably should be reproved in a (proofchecked) book at some point.
– YCor
Apr 10, 2020 at 15:23
• @YCor -- or even in a paper accepted by a journal.
– HJRW
Apr 12, 2020 at 15:41
• @HJRW My point being, written in a textbook style, rather than "original research" style, and complementary to the original paper.
– YCor
Apr 12, 2020 at 15:46