On groups generated by two positive multi-twists: Teichm"uller curves and Lehmer’s number

I’ll discuss a simple observation about a construction of
Thurston’s, from which we derive several interesting facts about subgroups
of the mapping class group generated by two positive multi-twists. In
particular, we identify all configurations of curves for which the
corresponding groups fail to be free, and show that a subset of these
determine the same set of Teichmuller curves as the non-obtuse lattice
triangles which were classified by Kenyon, Smillie, and Puchta. We also
identify a pseudo-Anosov automorphism whose dilatation is Lehmer’s number,
and show that this is minimal for the groups under consideration. In
addition, we describe a connection to work of McMullen on Coxeter groups
and related work of Hironaka on a construction of an interesting class of
fibered links.