Sigrid Böge, Moshe Jarden, Alexander Lubotzky
April 9, 2016

### 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 ,..., H n of infinite index such that G=⋃i=1nHiG${G=\bigcup _{i=1}^nH_i^G}$. Here, HiG={hg∣h∈Hi,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 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=⋃i=1∞HiG${G=\bigcup _{i=1}^\infty H_i^G}$, 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 .