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.