Geometric Interpretation of the Lower Central Series for the Fundamental Group?

Question

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.

Answer 1

At the risk of being obvious, and concerning Question 1:

What does the length (finite or infinite) of the chain tell us geometrically about X?

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).

Answer 2

There's Gromov's theorem, which almost gives a characterization of when the lower central series has finite termination in the trivial group when $G$ is finitely generated (giving a partial answer to the second part of Question 1).

Stallings has a nice application of lower central series to study maps between groups. This has been applied to get lower bounds on the growth of $p$-nilpotent subgroups under various conditions by Shalen-Wagreich and Lackenby. In particular, Shalen-Wagreich use the lower $p$-central series to show that a 3-manifold $M$ with $rank(H_1(M;Z_p))\geq 3$ has infinite fundamental group (although now this may be deduced from the geometrization theorem).

Examples where the lower $p$-central series can be computed come from analytic pro-$p$ groups. For a sample result, this was used by Boston and Ellenberg to show the existence of towers of 3-manifolds which are rational homology spheres (using examples of Calegari-Dunfield).

Addendum: Stallings' sequence with $N=G_1$, $Q=G/G_1=H_1(G)$ gives the exact sequence $$ H_2(G)\to H_2(G/G_1)\to G_1/G_2 \to 0$$ since $H_1(G)\to H_1(G/G_1)$ is an isomorphism. Notice that $G/G_2$ is a central extension of $G/G_1$ by $G_1/G_2$. To figure out which central extension, you take $H_2(G/G_1)/im\{H_2(G)\to H_2(G/G_1)\}$.

I think there's (sort of) a geometric interpretation of this, by dualizing. Consider $H^2(G/G_1;Q) \cong \wedge^2 H^1(G/G_1;Q)$ by taking cup products. If you consider the map $H^2(G/G_1;Q)\to H^2(G)$, then you see that the image is $\cup^2 H^1(G;Q)$. Dualizing, you see that the rank of the torsion-free part of $G_1/G_2$ is equal to the dim. of the kernel of the map $H^2(G/G_1;Q)\to H^2(G;Q)$, which is equal to $dim \wedge^2 H^1(G;Q)-dim \cup^2 H^1(G;Q)$.

Answer 3

A special case you might find informative: If $L$ is a link in $S^3$, then Chen-Milnor theory gives you a presentation of the link group $\pi=\pi_1(S^3-L)$ modulo some deeper terms of the the lower central series of $\pi$, and hence some information about some of the early lower central factors that you're asking about. Particularly neat is that this presentation is directly in terms of combinatorial invariants (linking number, Milnor invariants) of the link, and thus gives concrete interpretations to various cohomological invariants that arise from the algebraic topology viewpoint (cup product, Massey products, etc.)

Also worth mentioning is the gadget you get by gluing all of these lower central quotients together, namely the associated graded Lie algebra. (And you can repeat for the lower central p-series, and several other relevant series as well). There's a good amount known about these Lie algebras: For example, John Labute's "The Lie Algebra Associated to the Lower Central Series of a Link Group and Murasugi's Conjecture" (and the rest of Labute's paper for that matter. In particular, some amazing ties to "arithmetic topology" and number-theoretically interesting Galois groups).

Answer 4

As for question 4, the answer is well understood for the fundamental groups of surfaces, or more generally for "formal spaces" $X$. In summary: renumber your series so that $G_1=G$. Then $V_i:=G_i/G_{i+1}$ is a free $\mathbb Z$-module for all $i$, and the direct sum $L:=\bigoplus_{i\ge1}V_i$ is a graded Lie algebra over $\mathbb Z$. E.g. for surface groups, it admits as presentation the natural linearization of the group presentation: $$L=\langle A_1,B_1,\dots,A_g,B_g\mid [A_1,B_1]+\dots+[A_g,B_g]=0\rangle.$$ The enveloping algebra of $L$ is a quadratic Koszul algebra, and its Koszul dual $U(L)^!$ is isomorphic to the cohomology algebra of $X$. The ranks of the sections $V_i$ may be recovered from the Betti numbers of $X$ using Koszul duality and Möbius invertion.

More precisely, if $b_i$ are the Betti numbers of $X$, and $b(t)=\sum b_i t^i$ is its Poincaré series, and $c(t)=\sum c_i t^i$ is the Poincaré series of $U(L)$, then Koszul duality gives $c(-t)b(t)=1$. Then $c(t)=\prod_{i\ge1}(1-t^i)^{-\dim V_i}$ lets you compute $\dim V_i$.