On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number

From a simple observation about a construction of Thurston, 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 Teichmüller 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.