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|^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. 
Hessam Hamidi-Tehrani