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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
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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
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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 Answer 1

up vote 7 down vote accepted

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.

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Young has a second paper up: arxiv.org/abs/0912.2697 –  Qiaochu Yuan Dec 15 '09 at 15:56

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