Let be a group generated by two positive multi-twists. We give some sufficient conditions for to be free or have no “unexpectedly reducible” elements. For a group generated by two Dehn twists, we classify the elements in which are multi-twists. As a consequence we are able to list all the lantern-like relations in the mapping class groups. We classify groups generated by powers of two Dehn twists which are free, or have no “unexpectedly reducible” elements. In the end we pose similar problems for groups generated by powers of twists and give a partial result.
"Groups generated by positive multi-twists and the fake lantern problem." Algebr. Geom. Topol. 2 (2) 1155 - 1178, 2002. https://doi.org/10.2140/agt.2002.2.1155