Sliceable groups and towers of fields

Sigrid Böge, Moshe Jarden, Alexander Lubotzky

Research output: Contribution to journalArticlepeer-review


Let l be a prime number, K a finite extension of ℚl, 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,⋯, Hn of infinite index such that G = ∪ni=1 HGi. Here, HGi = {hg | h ∈ Hi, g ∈ G}. On the other hand, we prove for l ≠ 2 that no open subgroup of GL2(ℤl) is finitely sliceable and we give an arithmetic interpretation to this result, based on the possibility of realizing GL2(ℤl) as a Galois group over ℚ. Nevertheless, we prove that G = GL2(ℤl) has an infinite slicing, that is G = ∪i=1 HGi, 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.

Original languageEnglish
Pages (from-to)365-390
Number of pages26
JournalJournal of Group Theory
Issue number3
StatePublished - 1 May 2016

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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