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 D^×/K^× is finitely sliceable, i.e. G has finitely many closed subgroups H₁,⋯, H_n of infinite index such that G = ∪ⁿ_i=1 H^G_i. Here, H^G_i = {h^g | h ∈ H_i, g ∈ G}. On the other hand, we prove for l ≠ 2 that no open subgroup of GL₂(ℤ_l) is finitely sliceable and we give an arithmetic interpretation to this result, based on the possibility of realizing GL₂(ℤ_l) as a Galois group over ℚ. Nevertheless, we prove that G = GL₂(ℤ_l) has an infinite slicing, that is G = ∪^∞_i=1 H^G_i, where each H_i is a closed subgroup of G of infinite index and H_i ∩ H_j has infinite index in both H_i and H_j if i ≠ j.