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