We prove that every element of the mapping class group Γg has linear growth (confirming a conjecture of N. Ivanov) and that Γg is not boundedly generated. We also provide restrictions on linear representations of Γg and its finite index subgroups.
"Rank-1 phenomena for mapping class groups." Duke Math. J. 106 (3) 581 - 597, 15 February 2001. https://doi.org/10.1215/S0012-7094-01-10636-4