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The only normal generators of the Torelli group of a closed surface I can find is Powell's 1977 paper (where the presentation is a bit complicated and given essentially without proof). Is there any more enlightening or smaller presentation? I would also be interested in such a result for $\mbox{Aut}(\pi_1(F_g))$ (as opposed to $\mbox{Out}(\pi_1(F_g))$.

EDIT Actually, it seems that in the 1980 paper of Dennis Johnson he seems to show that Torelli is normally generated by a single element, which improves the Powell result.

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  • $\begingroup$ You might be able to extract a presentation from this paper: ams.org/mathscinet-getitem?mr=2552249 $\endgroup$
    – Ian Agol
    Commented Jun 4, 2012 at 18:49
  • $\begingroup$ @Agol : It's pretty hard to extract generators from that paper except when $g=2$. The trouble is that the simplicial cpx they are using (the "cycle cpx") has a pretty complicated action of Torelli, and it is hard to work out eg the quotient. However, the paper "Generating the Torelli group" by Hatcher-Margalit uses the cycle cpx to prove that a different cpx (defined by me in my paper "A note on the connectivity of certain complexes associated to surfaces") is connected, and from this they get generators for Torelli. I originally proved this connectivity using generators for Torelli! $\endgroup$
    – Andy Putman
    Commented Jun 4, 2012 at 21:17
  • $\begingroup$ @Andy: Ok, I see. Here's a link to their paper: arxiv.org/abs/1110.0876 $\endgroup$
    – Ian Agol
    Commented Jun 4, 2012 at 22:11
  • $\begingroup$ @Igor Rivin (in response to your edit) : Johnson's paper uses the theorem of Powell. What he does is give relations in Torelli which express Powell's generators in terms of a single normal generator (a "genus 1 bounding pair map"). By the way, this requires $g \geq 3$, though it is clear from Powell's work that in genus $2$ that Torelli is generated by the normal closure of a single Dehn twist about a separating curve. In fact, in his thesis Mess proved that the genus $2$ Torelli group is a free group on an infinite set of Dehn twists about separating curves. $\endgroup$
    – Andy Putman
    Commented Jun 6, 2012 at 2:42
  • $\begingroup$ @Andy: yes, but since my original question was asking about a smaller presentation, Johnson certainly provides it (I was asking about this because the Torelli subgroup for free group autos has one normal generator, and it seemed reasonable that the same would hold here...) $\endgroup$
    – Igor Rivin
    Commented Jun 6, 2012 at 4:05

1 Answer 1

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For the Torelli group of a surface, I gave a much easier and more geometric proof in my paper

MR2302503 (2008c:57049) Putman, Andrew(1-CHI) Cutting and pasting in the Torelli group. (English summary) Geom. Topol. 11 (2007), 829–865.

This proof was simplified quite a bit by Hatcher and Margalit in their paper "Generating the Torelli group" (to appear in L'Enseignement Mathématique).

For $\text{Aut}(F_n)$, modern proofs can be found in the paper

MR2336078 (2008k:57029) Bestvina, Mladen(1-UT); Bux, Kai-Uwe(1-VA); Margalit, Dan(1-UT) Dimension of the Torelli group for Out(Fn). (English summary) Invent. Math. 170 (2007), no. 1, 1–32.

and in my paper "The complex of partial bases for $F_n$ and finite generation of the Torelli subgroup of $Aut(F_n)$" with Matt Day.

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  • $\begingroup$ @Andy: Thanks! But (in case there is any misunderstanding), my second question is about $\mbox{Aut}$ of the surface group, not the free group... $\endgroup$
    – Igor Rivin
    Commented Jun 4, 2012 at 19:39
  • $\begingroup$ Whoops! Anyway, the automorphism group of a surface group is just the mapping class group of a surface with a single puncture. My paper give generators for the Torelli group of a surface with one boundary component, but this surjects onto the Torelli group of a punctured surface with kernel generated by the Dehn twist about the boundary component. This gives the desired result for punctured surfaces. $\endgroup$
    – Andy Putman
    Commented Jun 4, 2012 at 19:43

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