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.
Benson Farb. Alexander Lubotzky. Yair Minsky. "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