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

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.

• Possible duplicate of math.stackexchange.com/questions/142236/…
– lhf
May 15, 2018 at 10:37
• @lhf: Not at all ! May 15, 2018 at 10:39
• – lhf
May 15, 2018 at 10:41
• Just an aside: $L^{ab}$ as you've described it is a truly disgusting group. The correct definition of $L^{ab}$ for a topological group is $L/\overline{[L,L]}$, namely the quotient by the closure of the first derived subgroup. May 15, 2018 at 10:58
• Most importantly, infinite Galois theory associates extensions of $\mathbb Q$ with closed subgroups of $G_\mathbb Q$. If you take the closure of $[G_\mathbb Q, G_\mathbb Q]$, the resulting Galois group is, almost by definition, the Galois group of the maximal abelian extension of $\mathbb Q$. If you don't take the closure, the resulting group is meaningless from this perspective. May 15, 2018 at 11:06

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

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

• I guess there is no such nice description of $[L,L]$, or its closure ? May 15, 2018 at 12:59
• By Galois theory, $\overline{[L,L]}$ is $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q^{ab})$. There certainly isn't a nice description like there is for the quotient. This part of the Galois group really is the harder part to understand, and very little is known about it at all. May 15, 2018 at 14:31
• @Mathmo123: what is $\mathbb{Q}^{ab}$ ? Is anything known about the torsion of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}^{ab})$ ? May 17, 2018 at 9:56