Sliceable groups and towers of fields

Let l be a prime number, K a finite extension of Q(l), and D a finite-dimensional central division algebra over K. We prove that the profinite group G = D-x/K-x is finitely sliceable, i.e. G has finitely many closed subgroups H-1,..., H-n of infinite index such that G = boolean OR(n)(i=1) H-i(G). Here, H-i(G) = {h(g)vertical bar h is an element of H-i, g is an element of G}. On the other hand, we prove for l not equal 2 that no open subgroup of GL(2)(Z(l)) is finitely sliceable and we give an arithmetic interpretation to this result, based on the possibility of realizing GL(2)(Z(l)) as a Galois group over Q. Nevertheless, we prove that G = GL(2)(Z(l)) has an infinite slicing, that is G = boolean OR(infinity)(i=1) H-i(G), where each H-i is a closed subgroup of G of infinite index and H-i boolean AND H-j has infinite index in both H-i and H-j if i not equal j.