## New answers tagged mapping-class-groups

3
votes

Accepted

### Fixed points free automorphisms of Teichmüller spaces

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,...

3
votes

Accepted

### An interior cone condition for Teichmuller spaces

Many (most?) points of the boundary do not satisfy the interior cone condition. See the paper Spirals in the boundary of slices of quasi-Fuchsian space by Goodman.

1
vote

### Are the two families of Johnson invariants of the Torelli groups related beyond the first one?

Knowing $\text{gr}\ T$ only tells you about cup products and higher Massey products on $H^1(T;\mathbb{Q})$. Beyond that, there is no real relationship between $\text{gr}\ T$ and $H_{\ast}(T)$.
Your ...

5
votes

### Given a word $w$ in the braid group $B_n$, representing a pure braid, find the image of $w$ in the abelianization of $P_n$

There is a simple geometric way to think about the fact that $P_n^{\text{ab}} \cong \mathbb{Z}^{\binom{n}{2}}$: Draw the braid with strands going from left to right across the page. Pick any two of ...

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