# a question about mapping class groups

According to Thurston's construction, which can be found for instance in Farb-Margalit's A Primer on Mapping Class Groups, theorem 14.1 (here is a link to the version I am using: http://www.maths.ed.ac.uk/~aar/papers/farbmarg.pdf), If there are two curves $a_1$ and $a_2$ on a surface $S_{g,n}$ (a surface of genus $g$ with $n$ boundary components) that fill the surface, then we will be able to figure out the type of the element (periodic, reducible, Anosov) $\phi\in MCG(S_{g,n},\partial S_{g,n})$, $\phi=t_{a_2}t_{a_1}$, based on the intersection number of $a_1$ and $a_2$.

I want to know if anything can be said about the type of an element in the mapping class group that consists of three Dehn twists, say $\psi=t_{a_3} t_{a_2} t_{a_1}$, that $a_1$ and $a_2$ fill the surface and $t_{a_3}\notin <t_{a_1},t_{a_2}>$. Is there any general way we can determine whether such an element $\psi$ is periodic, reducible or Anosov (perhaps based on mutual intersections of these three curves)?

PS: I am actually working on a concrete example, so that $\psi=t_bt_at_c$, $\psi\in MCG(D_4,\partial D_4)$ ($D_4$ representing a $4$-puncutred disc) $a$ and $c$ fill the surface and $t_at_c$ is pseudo-Anosov, but $t_b\notin <t_a,t_c>$ as in the following figure:

• Do you have a link to the image? Jan 3 '18 at 3:48
• yes, this question: math.stackexchange.com/questions/2586232/…, but there $a$ and $c$ fill the surface, $t_at_c$ is Anasov and $t_b\notin <t_a,t_c>$. Jan 3 '18 at 4:40
• I've added it, please edit the question accordingly. If you want to insert a different image, temporarily insert it, say, on that math.se question and provide the link. Also if you have closely related questions on math.se and here, it is better to inform users here and there about it, to avoid effort duplications Jan 3 '18 at 4:45
• thanks, I edited the problem. I hope it is sufficiently clear now. Jan 3 '18 at 5:46
• Are you literally interested in words like $t_{a_2} t_{a_1}$ and $t_b t_a t_c$, where positive Dehn twists are indicated? Or are you interested more generally in allowing inverses as well, i.e. in allowing negative Dehn twists? I ask because, generally speaking, you must be very careful about signs of Dehn twists. Even if $a,c$ fill, you cannot in general expect the product of two positive Dehn twists like $t_a t_c$ to be (pseudo)-Anosov. On the torus, for instance, where $a,c$ represent the standard homology basis, the mapping class $t_a t_c$ has finite order whereas $t_a t_c^{-1}$ is Anosov. Jan 3 '18 at 20:14

If you are interested in a concrete example (as you suggest) then one approach is to use the computer programme flipper. Here is a link to the documentation.

I entered your example into flipper and found that the product $bac$ is pseudo-Anosov but the product $cab$ is reducible (it looks like it is pseudo-Anosov in a subsurface). When using flipper you'll want to check carefully that its conventions and yours match (for example, left versus right Dehn twists and left versus right actions of the mapping class group).

Regarding general triples: if you want to follow Thurston (and Veech) then you need to look for a singular euclidean structure on $S = S_{g,p}$ making all three of the Dehn twists into affine transformations (in fact, shears). Since Thurston is considering just a pair of twists, such a euclidean structure always exists. Since you are considering a triple of twists, then such a euclidean structure basically never exists. Still, you might get lucky...

Hmmm. But not in the example above, I think.

Not an answer, but worth $\epsilon$ more than a comment:

In general, if you have a pseudo-anosov $\phi$ (in your case, $t_a t_b$) and a Dehn twist $t_c$, then of the elements $\phi t_c^k$ at most six $k$ give rise to non-pseudo-Anosov elements, so if you were a betting man, I would bet on pseudo-Anosov (though granted, for a specific element, flipper is a better choice).

• Here Igor is referring to a theorem of Fathi. See Fathi's paper "Dehn twists and pseudo-Anosov diffeomorphisms". However, the bound proven there is seven, not six... I would be very interested to know the extremal examples here. Jan 3 '18 at 20:01
• Igor, $t_a t_b$ is not pseudo-Anosov - those curves only fill a four-holed subsurface of the five-holed sphere. Jan 3 '18 at 20:04
• @SamNead I was responding to the OP's original question (which made it sound like $t_a t_b$ was $\psi$A) - I did not look at the picture... Also the "six" makes what I cite the "Boyer-Zhang" theorem, "seven" was Fathi, "finitely many" - Long-Morton (but really Thurston). Jan 3 '18 at 21:27
• Aha! I found Boyer-Gordon-Zhang's paper "Dehn fillings of large hyperbolic 3-manifolds". I was not aware of this sharpening of Fathi's result - very nice. Thank you. Jan 4 '18 at 11:29