For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$G_0 > G_1 > … > G_i >…$
In the case where $G = \pi_1(X)$ The first quotient $H^1 = G_0/G_1$ is well known to be the first homology group (which has well known geometric content).
Question 1:
Are there geometric interpretations of further quotients $G_i/G_{i+1}$?
What does the length (finite or infinite) of the chain tell us geometrically about X?
We can also form the mod-p central series by taking $G^p_0 = G$, $G_{i+1}^p = (G_i^p)^p[G,G^p_{i}]$ and then again form the quotients $V_i^p = G^p_i/G^p_{i+1}$. In this case these are modules (vector spaces) over $Z_p$.
Question 2:
What are interpretations of these $V^p_i$? What can we say if we know their dimension (as a vector space) or if they're non-zero? I'm particularly interested in small i (= 1,2,3,4), and small p (= 2 say).
Question 3:
Are there good methods for calculating the $V_i^p$ (both direct and indirect)? For instance, a direct way would be to calculate them from a presentation of the fundamental group. Is this tractable (with software such as GAP) if the presentation is "small" in some sense? Can I bound their dimension (above or bellow)?
Are there indirect ways of calculating these vector spaces? As homology/cohomology of some other object on X? As something else? Group homology?
Question 4:
Is there a good source for these types of questions? Has somebody worked out the V's for compact surfaces (orientable or not)?
Answers or (even better) references to work on these types of questions would be great. I'm especially interested in examples worked out for surfaces and small i.
Best Answer
At the risk of being obvious, and concerning Question 1:
A group for which the lower central series terminates after finitely many steps is called nilpotent, and then the length of the lower central series is called the nilpotency class of the group.
There is a huge literature concerning how the nilpotency class of the fundamental group of a manifold interacts with its geometry. For a nice example, see
Belegradek, I.; Kapovitch, V. Pinching estimates for negatively curved manifolds with nilpotent fundamental groups. Geom. Funct. Anal. 15 (2005), no. 5, 929–938.
It's also worth mentioning that nilpotent spaces (spaces $X$ for which $\pi_1(X)$ is nilpotent and acts nilpotently on $\pi_k(X)$ for all $k\geq 2$) are an important class of spaces in homotopy theory. This is due to the fact that they admit "nice" Postnikov systems and as such behave well with respect to localization, making them amenable to the tools of rational homotopy theory (see this question and answers for more details).