Example/meaning of filtration on a group $(\mathbb{R},+)$

Solution 1:

Very good that you want to try out the definition on one of the standard examples! However, as people have pointed out in comments, it turns out that the group $(\mathbb R, +)$ just does not serve as a very motivational example for this concept.

A better example, or class of examples, would come from the upper triangular (unipotent) $n \times n$-matrices $G =\pmatrix{1&*&***&*&*\\0&1&***&*&*\\ &&\ddots&*&* \\0&0&0&1&*\\0&0&0&0&1}$ over any given field. The lower central series of this group is

$$G=G_1 \supsetneq G_2 = \pmatrix{1&0&***&*&*\\0&1&0**&*&*\\ &&\ddots&0&* \\0&0&0&1&0\\0&0&0&0&1} \supsetneq G_3 \supsetneq ... \supsetneq G_{n-1} = \pmatrix{1&0&0&0&*\\0&1&0&0&0\\ &&\ddots&0&0 \\0&0&0&1&0\\0&0&0&0&1} \supsetneq G_n = \{I_n\} =G_{n+1}=G_{n+2} = ...$$

and the corresponding filtration is $\omega(x) := \sup(n: x \in G_n)$. Indeed property 4 is equivalent to saying that the commutator $(G_i, G_j)$ is contained in $G_{i+j}$, a property which is sometimes called "being strongly central" and which the lower central series satisfies. Note that here, the filtration function takes only the finitely many values $\{1, ..., n\}$.

In fact, given a filtration $\omega$ with the specified properties on a group $G$ you get, for any $\nu \in \mathbb R_{> 0}$, subgroups $$G_\nu :=\{x \in G: \omega(x) \ge \nu\} \qquad \text{ and } \qquad G_{\nu^+} := \{x \in G: \omega(x) > \nu \}$$

satisfying $G = \bigcup_{\nu > 0} G_\nu$, and $(G_\nu, G_\mu) \subseteq G_{\nu+\mu}$, and $G_\nu = \bigcap_{\nu > \mu} G_\mu$. Conversely, if subgroups with these properties for all $\mu, \nu \in \mathbb R_{> 0}$ are given, they give back a filtration via the formula $\omega(x) = \sup(\nu: x \in G_\nu)$.

Note that for an abelian group, condition 4 is empty via condition 1, and basically we are just demanding a decreasing chain of subgroups.

Now it turns out that although the abelian group $(\mathbb R, +)$ indeed possesses tons of subgroup chains, none of them have attracted particular interest so far, probably because they shed not a strong light on anything anyone wants to know. But here's a filtration (I always mean the underlying additive group):

$$G_1= \mathbb R \supsetneq G_3 := \mathbb Q(\pi, \sqrt{6}) \supsetneq G_{12}:=\mathbb Q(\sqrt6) \supsetneq G_{4351}:=\mathbb Z \supsetneq G_{4352} :=17\mathbb Z \supsetneq G_{10000} := 85 \mathbb Z \supsetneq G_{20001}:=\{0\}$$

As a pointless exercise, you could write down the filtration function here explicitly, as a sum of indicator functions or something. As noticed by Sean Eberhard in a comment, in the case of $\mathbb R$ no continuous function can give out an interesting filtration.

One of the reasons one introduces such filtrations is that with them, we can define an associated graded object. Here, that object is

$$grG := \bigoplus_{\nu > 0} G_{\nu}/G_{\nu+}$$

which is checked to be a) an abelian group (i.e. $\mathbb Z$-module) on which b) one can define from the group commutator a bracket $[\cdot, \cdot]$ which indeed turns this into a (graded) Lie algebra (over $\mathbb Z$).

The functor $(G, \omega) \mapsto grG$ turns out to be, in some cases, an "algebraic" version of the classical Lie group - Lie algebra correspondence. And even if it's not, the "graded" object is often easier to study, but "many" properties which are easily checked on the graded object can then be translated back to the group.

Now if you use the filtration I jokingly proposed above for $\mathbb R$ and look at the associated graded object, it is a huge $\mathbb Z$-module with some more or less random torsion and trivial Lie bracket. Which really cannot shed much light on the structure of $\mathbb R$ itself (which happens to be a huge $\mathbb Z$-module, without torsion).

However, if you take the more serious example of those unipotent matrices, then you can check that the associated graded object is the Lie algebra of strictly upper triangular matrices, whose addition and Lie bracket are (arguably) easier to understand than the group $G$ in itself. (Note also that whatever field $K$ our group $G$ was defined over, this associated Lie algebra $grG$ is more than just a Lie algebra over $\mathbb Z$, it has a natural structure of a $K$-vector space.)

A very different example of a (this time abelian!) group with an interesting filtration is, for any discrete valuation ring $R$ with maximal ideal $\pi R$, the additive group $\pi R$ with $\omega(x)$ defined as the unique $n \in \mathbb N$ such that $x \in \pi^n R \setminus \pi^{n+1}R$.

For example, if you know the $p$-adic integers $\mathbb Z_p$ with their $p$-adic valuation $v_p$, the subgroup $p \mathbb Z_p$ naturally comes with a filtration just from restriction of that valuation, and the associated graded object is the polynomial ring $\mathbb F_p[x]$.

Another (abelian!) group of interest here is the multiplicative group of principal units $(1+p \mathbb Z_p, \cdot)$ with filtration $\omega(x) := v_p(x-1)$. This has the same associated graded object as the additive object above (and this has somehow to do with the exp-log correspondence).

And now, one of the coolest things I have ever seen in math is what happens when you throw the idea of algebraic or Lie groups from above together with $p$-adic structures. Michel Lazard wrote a most beautiful monograph about that, and it all starts with these filtrations.