  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 Teichmueller 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.
