Sliceable groups and towers of fields

Abstract

Abstract 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 × / K × ${G=D^\times /K^\times }$ is finitely sliceable, i.e. G has finitely many closed subgroups H 1,...,Hn of infinite index such that G = ⋃ i = 1 n H i G ${G=\bigcup _{i=1}^nH_i^G}$ . Here, H i G = { h g ∣ h ∈ H i , g ∈ G } ${H_i^G=\lbrace h^g\mid h\in H_i, \, g\in G\rbrace }$ . 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 i G ${G=\bigcup _{i=1}^\infty H_i^G}$ , where each Hi is a closed subgroup of G of infinite index and Hi ∩ Hj has infinite index in both Hi and Hj if i ≠ j.