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Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\mathbb{C}^N$ via the Bers embedding, the mapping class group acts on $\mathcal{T}_{g,n}$ as holomorphic automorphisms of $\mathcal{T}_{g,n}$.

Is there always a fixed points free automorphism of $\mathcal{T}_{g,n}$?

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    $\begingroup$ A mapping class fixes a point in Teichmuller space precisely when it is an automorphism of the corresponding Riemann surface, and thus has finite order. $\endgroup$
    – Andy Putman
    Commented Feb 17 at 20:56

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Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one, depending on your background.) A pseudo-Anosov homeomorphism induces a fixed-point-free isomorphism of the Teichmüller space.

Edit: As Andy points out, any Dehn twist (about an essential simple closed curve) also gives an example.

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    $\begingroup$ There is no need for a pseudo Anosov. A Dehn twist also has no fixed points. Pseudo Anosovs are needed for the stronger property of having positive translation distance. $\endgroup$
    – Andy Putman
    Commented Feb 17 at 21:42
  • $\begingroup$ Many apologies for asking such a trivial question. I was thinking about a generalization of Cartan's theorem for Teichmuller spaces. The classical Cartan theorem asserts that a bounded domain $\Omega \subset \mathbb{C}^k$ has a noncompact automorphism group if and only if there are a point $x \in \Omega$, a point $p \in \partial \Omega$ and a sequence $\{\phi_i\}_i$ of automorphisms of $\Omega$ such that $\lim_{i \rightarrow \infty} \phi_i(x)=p$. Now the question is the following: Can we choose the sequence of automorphisms as the iterates of a single automorphism? $\endgroup$
    – Mahdi Teymuri Garakani
    Commented Feb 18 at 1:40
  • $\begingroup$ I was thinking from the several complex variables viewpoint. I completely forgot about the mapping class group side. $\endgroup$
    – Mahdi Teymuri Garakani
    Commented Feb 18 at 1:52
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    $\begingroup$ @MahdiTeymuriGarakani - well, in the case of Teichmüller space, the answer is “yes you can use iterates”. I can’t help you with the general case. Sorry! $\endgroup$
    – Sam Nead
    Commented Feb 18 at 19:15
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    $\begingroup$ @LSpice - Thank you for the edition. $\endgroup$
    – Mahdi Teymuri Garakani
    Commented Feb 18 at 21:08

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