Let Γ be a group generated by two positive multi-twists. We give some sufficient conditions for Γ to be free or have no "unexpectedly reducible" elements. For a group Γ generated by two Dehn twists, we classify the elements in Γ 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 n ≥ 3 twists and give a partial result.

Formanek and Procesi have demonstrated that Aut(F n ) is not linear for n ≥ 3. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group, in Aut(F n ), from which they build an FP-group. We first prove that poison groups cannot be embedded in certain mapping class groups. We then show that no FP-groups of any form can be embedded in mapping class groups. Thus the methods of Formanek and Procesi fail in the case of mapping class groups, providing strong evidence that mapping class groups may in fact be linear.

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We prove that the word problem in the mapping class group of the once-punctured surface of genus g has complexity O(|w| 2 g) for |w| ≥ log(g) where |w| is the length of the word in a (standard) set of generators. The corresponding bound in the case of the closed surface is O(|w| 2 g 2 ). We also carry out the same methods for the braid groups, and show that this gives a bound which improves the best known bound in this case; namely, the complexity of the word problem in the n−braid group is O(|w| 2 n), for |w| ≥ log n. We state a similar result for mapping class groups of surfaces with several punctures.

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