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.



For which surfaces is Penner's conjecture known to be true?
Shin and Strenner have shown that the conjecture is false when 3g + n > 4. 


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


