TY - JOUR
AU - Lubotzky, Alexander
AB - 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 = 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 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 ∞ H i G ${G=\bigcup _{i=1}^\infty H_i^G}$ , 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 .
TI - Sliceable groups and towers of fields
JF - Journal of Group Theory
DO - 10.1515/jgth-2016-0509
DA - 2016-05-01
UR - https://www.deepdyve.com/lp/de-gruyter/sliceable-groups-and-towers-of-fields-Jbs2USIeDo
SP - 365
VL - 19
IS - 3
DP - DeepDyve
ER -