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Suppose $X$ is an orientable surface with non-empty boundary and $f:X\to X$ is a pseudo-Anosov automorphism that acts identically on $H_1(X,\mathbf{Z})$. Let $x$ be a fixed point of $f$.

For any $\gamma\in\pi_1(X,x)$ we have $\gamma^{-1}f(\gamma)\in [\pi_1(X,x),\pi_1(X,x)]$, the commutant of $\pi_1(X,x)$. More generally, we have $\gamma\cdot g^{-1}f g(\gamma)\in [\pi_1(X,x),\pi_1(X,x)]$ where $g$ is an automorphism of $X$ that fixes $x$.

I would like to ask what one can say about the normal closure in $\pi_1(X,x)$ of the set of all elements $\gamma\cdot g^{-1}f g(\gamma)$ where $\gamma$ runs through $\pi_1(X,x)$ and $g$ runs through the set of all diffeomorphisms $X\to X$ that fix $x$. In particular, does this closure coincide with the commutant of $\pi_1(X,x)$?

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No. Let $\Gamma_i$ be the lower central series defined by $\Gamma_1=\pi_1(X,x)$, $\Gamma_{i+1}=[\Gamma_1,\Gamma_i]$. The Johnson filtration $\text{Mod}_g(k)$ is the descending filtration of the mapping class group relative to $x$ defined by:

$f\in \text{Mod}_g(k)\iff f$ acts trivially on $\Gamma_1/\Gamma_k$

The first term $\text{Mod}_g(2)$ is the Torelli group, consisting of diffeomorphisms acting trivially on homology. The next term $\text{Mod}_g(3)$ is the Johnson kernel. By a beautiful theorem of Johnson, this is the subgroup generated by Dehn twists around separating curves.

By residual nilpotence of surface groups, we have $\bigcap \text{Mod}_g(k)=\{1\}$, but every individual term in the filtration is nontrivial. It is not hard to see that every term of the Johnson filtration contains pseudo-Anosovs. Indeed every normal subgroup of the mapping class group contains pseudo-Anosovs (see Lemma 2.5 of Long, "A note on the normal subgroups of mapping class groups") from which Long concluded that any two normal subgroups intersect nontrivially!

Thus since $\text{Mod}_g(k)$ is normal, if we take $f\in\text{Mod}_g(k)$ we have $\gamma^{-1}\cdot g^{-1}fg(\gamma)\in \Gamma_k$ for all $g$ and all $\gamma$.

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  • $\begingroup$ Thanks, Tom! By the way, could you give me a reference for the fact that every normal subgroup of the mpg contains a pseudo-Anosov map? $\endgroup$
    – algori
    Commented May 8, 2010 at 23:32
  • $\begingroup$ I added a reference above. There is one approach I like: if f in N is reducible, conjugate it to some g with a totally different reduction system; then combining big powers of f and g should give pseudos as in Thurston's argument. But I do not know if anyone has ever made this work. Long instead plays ping-pong with pseudos on PL (the space of projective laminations). He gets for example that for any normal N, the set in PL of fixed points of pseudos in N is dense in PL. And as he points out, you don't need so much to make ping-pong work -- you can take N normal only in Torelli, etc. $\endgroup$
    – Tom Church
    Commented May 8, 2010 at 23:59

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