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Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along the curves $a_i$s, and negative powers of Dehn twists along curves the $b_j$s, such that each curve in $A\cup B$ appears at least once, is a pseudo-Anosov homeomorphism of the surface $S$ (see http://www.jstor.org/stable/2001116).

He also conjectures that any pseudo-Anosov mapping class has a power which is representable as a composition of Dehn twists as above.

This conjecture is true for the surfaces $S_{1,1}$ and $S_{0,4}$, essentially as a consequence of the fact that the curve complex of these two surfaces embeds in their Teichmuller space ($\mathbb{H}^2$) through the Farey tessellation, and translations of the triangles of the Farey tessellation are representable as Dehn twists.

I was wondering if there are other surfaces for which the conjecture is known to be true, and where to find a proof.

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2 Answers 2

up vote 8 down vote accepted

The conjecture is not known for any other surfaces.

I'll throw out that there are other known recipes for constructing all pseudo-Anosov mapping classes using train track representatives; see for example my paper "The classification of pseudo-Anosovs", and the Bestvina-Handel paper "Train-tracks for surface homeomorphisms". However, none of these can be regarded as positive evidence for Penner's conjecture.

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Shin and Strenner have shown that the conjecture is false when 3g + n > 4.

See http://arxiv.org/abs/1410.6974

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1  
That's a nice development. –  Ryan Budney Nov 4 at 17:25

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