Abstract properties of the absolute Galois group over $\mathbb{Q}$

Question

Let $L := \mathrm{Gal}\Big(\overline{\mathbb{Q}}/\mathbb{Q}\Big)$ be the absolute group over the rationals. What are the most important and/or interesting abstract group theoretical properties of $L$ and $L^{ab} := L/[L,L]$, where $[L,L]$ is the first derived subgroup of $L$ (that is, the commutator subgroup of $L$) ? For instance, what is known about torsion in both $L$ and $L^{ab}$ ? Are $L$ or $L^{ab}$ (related to) free groups ?

Just to be clear: I am not asking about information about the various interesting actions of $L$ or $L^{ab}$ on geometric objects; rather, I am wondering about abstract group theoretical properties.

Answer 1

The Kronecker–Weber theorem describes the maximal abelian extension of $\mathbb Q$: it's the union of all cyclotomic extensions. Moreover, $L^{ab} \cong \prod _p \mathbb Z_p^\times$.

Answer 2

Here are a few precarious leads. I denote by $G=G_{\mathbf Q}$ the absolute Galois group of $\mathbf Q$ (I definitely cannot vote for $L$ !), $G^{ab}$ its maximal abelian quotient. Let us warn once and for all that all infinite Galois groups are profinite groups (= proj. lim. of finite groups), and subgroups will mean closed (closure of) subgroups. Since $G^{ab}=G/[G, G]$ it is natural to "approximate" $G$ by the series of successive quotients $G^{[n]}=G/G^{(n)}, n=1, 2, ...$, where $G^{(n)}$ is the derived series defined by $G^{(1)}=G,...,G^{(n+1)}=[G^{(n)},G^{(n)}]$, or preferably the descending central series defined by $G^{(n+1)}=[G,G^{(n)}]$, or (for technical reasons, see below), the series $G^{(n+1)}=G^2[G,G^{(n)}]$. Let us concentrate on the latter case, so $G^{[2]}=$ the maximal abelian quotient of exponent $2$ $ \cong Hom (\mathbf Q^*/{\mathbf Q^*}^2, (\pm 1))$ (by Kummer theory), and $G^{[3]}= $ (??)

Until recently, $G^{[3]}$ was not even known explicitely ! A first step was to find a canonical $\mathbf F_2$-basis of $({({\mathbf Q_{ab}}^*/({\mathbf Q_{ab}}^*)^2)}^{G^{ab}}= Hom (G^{(3)},(\pm 1))$. This was done by Anderson using CFT (2002): let $A$ be the free abelian group on the symbols $[a] \in\mathbf Q/\mathbf Z$ and let $sin : A \to {\mathbf Q_{ab}}^*$ the unique homomorphism s.t. $sin [a]=2sin (\pi a)$ for $0<a<1$, $0$ for $a=0$. The desired basis consists of the classes mod $({\mathbf Q_{ab}}^*)^2$ of {$\sqrt l$}$ \cup${$sin (a_{r,q}$)}, where $l$ runs over the prime numbers, $(r,q)$ runs over all the pairs of primes $r<q$, and $a_{r,q}$ is a combinatorial expression too complicated to be reproduced here (see the main thm. of [A]). But this is not enough, what is wanted is a Galois description of $G^{[3]}$, which was done by Efrat & Minac (2011) using techniques of the embedding problem: the fixed field of $G^{(3)}$ is the compositum over $\mathbf Q$ of all the normal extensions of $\mathbf Q$ with Galois groups $C_2, C_4$ (cyclic) and $D_8$ (dihedral). To my knowledge, this is the state of the art in this kind of approach.

NB: This last part is a purely algebraic result, valid for any field of characteristic $\neq 2$. Actually,the result of Efrat-Minac can be extended to any field of characteristic $\neq p$, containing a primitive root of unity of order $q=p^a$ (replace $2$ by $q$).

[A] G. W. Anderson, Kronecker-Weber plus epsilon, Duke Math. J. 114, 3 (2002), 439-475