# Quotient groups of the lower central series of a surface group

In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.

Q. What are the quotient groups of the lower central series of the genus $g$ surface group $\Pi_g$, namely $$\Pi_g = \langle a_1, \ldots, a_g, \, b_1, \ldots, b_g \; | \; [a_1, \, b_1] \cdots [a_g, \, b_g]=1 \rangle?$$

I guess that it is a very classical subject with a huge literature, but I am not an expert in the area, so any pointer to some relevant book/article will be higly appreciated.

• A description of the subquotients for surface groups is given in an answer to this MO-question: mathoverflow.net/questions/87160 – Matthias Wendt Jun 7 '18 at 9:52
• I see, thanks. A sketch for the computation of the quotient groups is given in the last answer, but it is not completely clear to me, and there is no reference given. By any chance, do you have one? – Francesco Polizzi Jun 7 '18 at 9:57

This paper seems to be relevant:

• S. Papadima and S. Yuzvinsky. On rational $K[π,1]$ spaces and Koszul algebras. J. Pure Appl. Algebra 144 (1999), no. 2, 157–167. (link to paper on ScienceDirect)

At least it contains the formula for dimenions in terms of Betti numbers mentioned in grok's answer to the MO-question I pointed out in the comments. There is a significant amount of further literature related to associated Lie algebras, Malcev completion of fundamental groups, Koszul duality properties and Chen's iterated integral. (You can either start browsing using these keywords, or start with papers coauthored by Papadima or Suciu and work through the references. I could also provide some more references if you state more precisely what type of answer you are looking for.)

Maybe on a historic note, it seems that the subquotients for the lower central series of one-relator groups were first computed by Labute (but this doesn't involve the Koszul point of view, rather the relation to the lower central series for free groups).

• J.P. Labute. On the descending central series of groups with a single defining relation. J. Algebra 14 1970 16–23.
• Nice. The first paper is a good introduction to this Koszul algebra stuff, whereas the computation at page 19 of the second one helped me to understand why Moebius inversion comes into play. Thanks! – Francesco Polizzi Jun 7 '18 at 10:46
• I'm really not an expert on this Koszul algebras theory, so let me ask something (probably) trivial. Is a genuine $K(\pi, 1)$-space a rational $K[\pi, 1]$-space in the sense of Papadima and Yuzvinsky? For instance, is the product of a finite number of smooth projective curves of genus at least $2$ a rational $K[\pi, 1]$-space? – Francesco Polizzi Jun 8 '18 at 13:37
• @FrancescoPolizzi: I would expect so, but need to check. My favourite references for rational homotopy usually assume that the spaces in question are nilpotent. – Matthias Wendt Jun 11 '18 at 15:05
• @FrancescoPolizzi: concerning the explicit question: yes the product of curves is a rational $K[\pi,1]$ in this sense. First, we check that the cohomology of a curve is Koszul: it is generated by $a_1,\dots,a_g, b_1,\dots, b_g$, and the relations are that $a_i\cup b_j=-b_j\cup a_i$ is always the same generator of $H^2$ if $i=j$ and 0 otherwise, all other cup products are trivial. Then the tensor product of Koszul algebras is Koszul. So the product of curves is a formal space whose cohomology is Koszul, hence it is a $K[\pi,1]$. – Matthias Wendt Jun 11 '18 at 15:08
• In response to a question of Francesco Polizzi from above: no, a $K(\pi,1)$-space need not be a rational $K(\pi,1)$-space. For instance, the complement of the reflection arrangement of type ${\rm D}_n$ is aspherical (and also formal), but it is not a rational $K(\pi,1)$ if $n\ge 4$. – Alex Suciu Jul 15 '18 at 21:29