On complexity of the word problem in braid groups and mapping class groups

We prove that the word problem in the mapping class group of the once-punctured surface of genus g has complexity O (|w|2g) 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|2g2) . 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|2n) , for |w|≥logn . We state a similar result for mapping class groups of surfaces with several punctures.