# The action of torsion of $MCG(S)$ on curve complex

Hi everyone.

Let $S$ be a closed surface with genus at least 3, $\alpha, \beta$ be the two vertices of curve complex of $S$ such that $d_{\mathcal {C}(S)}(\alpha, \beta)\geq 3$.

My question is

Is there a non-trivial finite ordered element $f$ of $MCG(S)$ such that $f(\alpha)=\alpha$ and $f(\beta)=\beta$ in $\mathcal {C}(S)$?

Thanks!

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Are you asking whether $f$ exists given $\alpha, \beta$, or whether there exists $\alpha,\beta,f$ with that property? –  Ian Agol Dec 14 '12 at 4:21
Either way, the answer is yes, with $f=Id.$ –  Igor Rivin Dec 14 '12 at 4:56
@Igor Rivin : Of course, that's not what they OP is looking for. And it's not a silly question (at least under Agol's second interpretation) -- there definitely exist nontrivial finite order mapping classes which fix a filling pair of curves (they permute the discs in the complement). –  Andy Putman Dec 14 '12 at 5:17
@Agol, the last one. –  yanqing Dec 14 '12 at 6:11
Actually it's not silly under Agol's 1st interpretation either. Existence of $\alpha,\beta$ for which there is no such $f$ is interesting to ponder. –  Lee Mosher Dec 14 '12 at 13:58

Here is a way to find lots of examples. Suppose that $\Sigma$ is a surface and suppose that $f$ is a periodic mapping class. Let $S$ be the quotient orbifold $\Sigma/f$. Then taking full preimages gives a quasi-isometric embedding of the curve complex of $S$ into the curve complex of $\Sigma$. See

arXiv:1104.3492 and arXiv:math/0701719

for two different proofs. When $S$ has an infinite diameter curve complex we can pick $a$ and $b$, curves in $S$, that are as well separated as you want. Then the preimages $\alpha$ and $\beta$ are also well separated, and fixed as simplices by $f$. A bit more work (which I haven't done) should give examples where $\alpha$ and $\beta$ are single vertices. There are also examples where $S$ is honestly a surface, not just an orbifold.

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Well done, thanks! –  yanqing Dec 14 '12 at 12:11
Here is a way to get a pair of curves rather than pair of multi-curves: Let $S$ be a surface of genus $\ge 2$, $c\in H_1(S,Z_n)$ a nontrivial primitive class, $a, b$ simple curves representing this class. By applying powers of pseudo-Anosov in the Torelli subgroup to $b$, we can assume that $dist(a,b)$ is large. Now, take a homomorphism $f:\pi_1(S)\to Z_n$ sending $c$ to the generator and let $\Sigma$ be the cyclic cover of $S$ corresponding to the kernel of $f$. Then lifts of $a, b$ to $\Sigma$ are connected. Now, proceed as in Sam's answer. –  Misha Dec 14 '12 at 17:27
@Misha. It is great. Thanks again! –  yanqing Dec 15 '12 at 2:06

Here is an example which can be easily generalized for all $S_g$, $g\geq 3$. With similar approaches, one can construct many examples, not just starting with the building block $S_{1,2}$. Notice the Dehn twist about one boundary component in $S_{1,2}$.

http://s11.postimage.org/ajzyt2xsz/finiteorderfill.png

Given a finite group $G$ of mapping classes, can one find $S_g$, and curves $a,b$ on $S_g$ that fill such that $a,b$ are stabilized by each element of $G$? (off the top of my head, the example I have given does each finite cyclic group of order at least 3)

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If you look closely at Sam's construction and my comments, you will see that it gives examples for all genera $\ge 3$ since every such surface is a cyclic cover of genus 2 surface. –  Misha Dec 19 '12 at 2:31
You are correct, thank you for your comment. In case yanqing wanted an explicit example, yanqing now has one. Having explicit examples can be beneficial. –  user30022 Dec 19 '12 at 11:59
great, thank you for your explicit example. –  yanqing Dec 19 '12 at 14:24