# How do Dehn functions of special linear and mapping class groups behave?

Hi, I apologize for the basic questions. I am looking for good references on the following problems:

1) What is known about the Dehn function of $SL_n(\mathbb{Z})$?

2) What is known about the Dehn function of mapping class groups?

Thanks!!

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is like a generalization of a Dehn twist in the 2-torus? – janmarqz Dec 8 '09 at 19:04
No, it is a measure of how hard it is to "fill discs" in the group. See Martin Bridson's beautiful survey "The geometry of the word problem" for a discussion of this. – Andy Putman Dec 8 '09 at 19:49
It might be useful to change to title so that it mentions the special linear group and dthe mapping class group. – Andy Putman Dec 8 '09 at 20:04
As motivation for people reading this, the "Dehn function" for a group is also often called the "isoperimetric function" for the group. – Andy Putman Dec 9 '09 at 2:05

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.