gr.group theory - How do Dehn functions of special linear and mapping class groups behave?

1) For $SL_n(mathbb{Z})$, it depends on the dimension.

a. For $n=2$, the group is virtually free, so the Dehn function is linear.

b. For $n=3$, a theorem of Thurston-Epstein says that it is exponential (this can be found in the book "Word Processing in Groups").

c. For $n>3$, Thurston conjectured that it should be quadratic. This was recently proven for $n>4$ by Robert Young. He hasn't yet written up the proof, but he has a preprint here proving that it is at most quartic.

2) For the mapping class group, Lee Mosher proved that it is automatic (see his paper "Mapping class groups are automatic" in the Annals in 1995; a survey of the proof can be found in his beautiful paper "A user's guide to the mapping class group: once-punctured surfaces"). As shown in "Word processing in groups", this implies that its Dehn function is quadratic.