Relation between $G_{\mathbb{Q}_p}$ for different primes

Question

Let $G_{\mathbb{Q}_p}$ denote the absolute Galois group of the $p$-adic field $\mathbb{Q}_{p}$. Also, their structure as abstract groups is completely known.

It is well known that this group embeds into the absolute Galois group of the rationals, $G_{\mathbb{Q}}$.

My question is: are there any known relations between the two absolute Galois groups of two $p$-adic fields, say $\mathbb{Q}_{p_1}$ and $\mathbb{Q}_{p_2}$, where $p_1$ and $p_2$ are different primes, regarded as subgroups of the absolute Galois group of the rationals?

Answer 1

As noted in the remarks above, there are different (although conjugate) embeddings $G_{\mathbb{Q}_p}\rightarrow G_\mathbb{Q}$. Let us fix one of them and denote its image by $G_p$.

If we fix prime numbers $p_1,\dots,p_n$ (not necessarily distinct), then the set of $\sigma=(\sigma_1,\dots,\sigma_n)\in G_\mathbb{Q}^n$ for which the closed subgroup $\left< G_{p_1}^{\sigma_1},\dots,G_{p_n}^{\sigma_n}\right>$ is the profinite free product of $G_{p_1},\dots,G_{p_n}$ has Haar measure $1$. This is (a special case of) Theorem 4.1 in W.-D. Geyer, Galois groups of intersection of local fields, Israel Journal of Mathematics 30, 1978.

Geyer also notes in section 4.4 that one cannot expect this to hold for every $\sigma$, but his example is for archimedean places. A purely group theoretic result of Haran occurring as Proposition 4.2.3 in Neukirch-Schmidt-Wingberg's Cohomology of Number Fields however shows that also for non-archimedean places it does not hold for every $\sigma$.