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We study subgroups of the mapping class group of the torus generated bypowers generated by powers of Dehn twists. We give a criterion to show when acollection of powers Dehn twists generates a free group using the ping ponglemma. We show that the subgroup generated by three uniform powers of Dehntwists can be either the whole mapping class group, a direct product of a freegroup of rank two with the cyclic group of order two, or a free group of rankat most three. We characterize subgroups generated by a collection of(respectively squares of) Dehn twists, if there is a pair among so that thegeometric intersection number of the corresponding simple closed curves is two(respectively one). We also determine the subgroup generated by uniform powersof all Dehn twists.

Subgroups of the mapping class group of the torus generated by powers of Dehn twists